The three OVAs of Gunsmith Cats, in addition to telling their own storyline rather than risking trying to adopt the manga storyline, also toned a lot of things down. May Hopkin's status as an underaged child prostitute and her sexual fetish for explosions and the scent of gunpowder, Rally Vincent's tendency to stop her opponents by blowing off their fingers or arms, the firepower of May's handgrenades, the general copious amounts of death, all of it went out the window. It is still pretty violent, though.
Pokémon is considerably lighter and softer than the games, which are slowly growing darker and darker as they go on. Some dark elements still show up on occasion though, especially in the movies. In the show itself, they just don't stick very long.
Example A: Team Galactic, a very dark, sinister villain group from the video game, show up in the Diamond/Pearl series....in just 10 episodes out of 191.
Example B: Due to trauma stasis, the two episodes involving Team Plasma fighting Team Rocket and creating mass destruction in Castelia City were not aired, and the Team was introduced in a different way with little to no themes based on the video games essentially turning them into generic villains.
The anime adaptation of the original Slayers novels has its fair share of emotional turmoil and grisly moments (enough to be denied the right to air on the old Fox Kids network back in the late 90's, believe it or not), but in general it is much lighter in tone and sillier (replacing the female protagonist's First-Person Smartass humor from the novels to slapstick). Also, the character Xellos, an Affably Evil demon-priest, is far more sadistic in the novels, whereas his affably-evil attitude in the anime makes him comical. A story involving a cult supporting the world's Big Bad was also never bought into anime form.
Most of the manga and video games that followed are also this trope.
Sailor Moon played this straight for Pretty Guardian Sailor Moon. Some elements were toned down in the anime compared to the manga, but it wasn't lighter. For instance, in the first story arc, only Usagi dies (from using the crystal's full power) and Mamoru revives her with a kiss; whereas the anime version had everyone brutally die in pure Nightmare Fuel. Then again, Usagi was less of a "bunny" in the anime version.
Also done in the anime itself with the SuperS season, after the previously unseen level of darkness in the preceding S season.
It's still fairly dark, though. That it manages to qualify as lighter and softer says far more about the original than it.
Girlfriend of Steel does this too, obviously. Everyone seems to be alive and flourishing. Rei is a Genki Girl. Yikes. Oddly enough despite that Rei has the exact same backstory.
Parallel Trouble Adventure Dual takes the general characters — a brown-haired boy with parental issues, a complex soul tie to a giant mecha, and rare hallucinations, a tsundere girl trying to prove herself, and a partially alien 'soul' inhabiting an artificial body with the resulting identity and self-worth issues — and setting of Neon Genesis Evangelion and looks at how it'd perform if everyone involved was at least sane enough to have not taken a pencil to their eyeballs long ago. Given that it belongs to the Tenchi Muyo!Multiverse it's hardly surprising.
Argento Soma, another Take That series, is still fairly grim, and its heroes are still full of issues, but overall seem to cope pretty well, up to making the half of the cast Warrior Therapists, as the authors believed that being badass makes wonders with one's mental health.
Probably the best example in EVA spin-off, is Evangelion @ School (or Petit Eva), where there are chibis of MASS-PRODUCED EVAS!
The anime version of Magic Knight Rayearth does this with Alcyone's death. Not only does Alcyone not die after the first failed attempt at killing the Magic Knights, but she actually gets to redeem herself and save Cephiro in the second season.
Subverted when Presea is actually killed off. In the manga she never dies though so naturally the anime fixes that.
Axis Powers Hetalia does this for all of history. The anime took it even further. But it still applies to the comic itself; the earlier strips had a lot more dark humour.
Most anime adaptations of Osamu Tezuka's manga get this, as most of the great man's work, while having a generally cartoony look & feel with an uplifting message, still involved copious amounts of death and destruction (Lost World, one of his earliest forays into the world of boys' adventure stories was infamous at the time for having the highest body count of any manga up to that point, with only three members of the cast surviving to the end of the book). In more recent years this has started to swing the other way, though (the Metropolis animé is far more bloody than the original, and the 2003 Astro Boy series' version of Dr. Tenma takes the Evilutionary Biologist gimmick that was only briefly touched upon in the manga and turns it up to 11).
In light of this, it is interesting to consider Astro Boy: Omega Factor for the GBA. The first half of the game shows the world growing steadily Darker and Edgier; the second half is Astro making things lighter and softer.
At some points, things just get weird ( OK, so Eponine and Enjorlas still have to die, yet Javert, the freaking antagonist, DOESN'T commit suicide and LIVES?)
The Melancholy of Haruhi-chan Suzumiya, a web series spin off of a similarly titled blockbuster franchise, seems very much a lighter and softer take on its source material, rendering every character as a Super-Deformed version, recasting Yuki Nagato as an eroge-obsessed otaku, and generally coming across as the original series' writers, animators and voice actors just having fun with the source material.
Full Metal Panic? Fumoffu. The primary mech is a ridiculously cute mascot character named Bonta-kun, which can only "speak" by squeaking a high-pitched "Fumoffu!" Sandwiched between the original, serious series and the Darker and Edgier The Second Raid, it makes the differences all the more pronounced.
The fluffy Bonta-kun suit is also quite literally both lighter and softer than the huge metal mechas from the original and "The Second Raid."
Even in The Second Raid, Sousuke shows his trust towards Kaname by letting her cut his hair. Let's review it: a teenage Child Soldier conditioned to perceive everyone as a potential enemy is letting another person handle scissors near his head and behind his back (falling asleep in the process). More so, he actually gets aroused by her proximity even though he is The Stoic. It doesn't make him any less of a Bad Ass, though. That's just some good character development.
Death Note certainly had more comedic moments while Light was no longer Kira as the result of a Memory Gambit, mostly because of Misa and Matsuda. Compared to the rest of the series only one named character was killed and he was a minor antagonist, and it was offscreen and the Kira Investigation Unit's base gave everything a sort of sitcom vibe.
SD Gundam Force. It's Gundam, but all the characters are chibis, everything is completely light-hearted, and hardly anyone ever dies. Though it can have it's nightmare fuel-like moments. That said, it's only slightly lighter and softer than some of the SDGundam stuff that preceded it...
 is decidedly less (visibly) brutal than many of Tomino's other Gundam works, with more lighthearted and very optimistic (or at least hopeful) themes.
Stitch!, the anime adaptation of Lilo & Stitch and Lilo & Stitch: The Series, is not just more lighthearted than the originals, but also replaced the human characters, and is possibly an alternate version of the series entirely.
In the original H-Game series, there was only one penalty for losing which applied to all matches: the victor was asked to leave the ring, while their opponent was publicly raped by members of the audience, or members of the tournament committee.
In the original Yu-Gi-Oh!, Yugi once had someone who had a gun to his face set himself on fire and kill himself. Joey (Jonouchi) was involved in a vicious street gang, that beat the crap out of him, and almost murdered him and Yugi. Tea (Anzu) was almost blown up by a terrorist. People got into vicious fights. People got offed now and then. In later adaptations of the franchise, everybody just plays cards.
Yu-Gi-Oh! GX is lighter and softer than its predecessor up until season 3.
Yu Gi Oh 5 DS, which had a gritty apocalypse story, was followed by Yu-Gi-Oh! ZEXAL, which is a much more optimistic series with a space-age theme and more focus on comedy.
Not quite to the same degree, but the manga adaptation of Fate/stay night itself is leaning in this direction. It was already focusing mostly on the 'Fate' route of the visual novel, which is notable for being the most idealistic of the three routes, but then went even further by removing the multiple rapes and other trauma in Sakura's backstory, and humanizing and sparing the life of Shinji.
As both series are intended to be comedies to the core, it would be inaccurate to claim that either version of Ranma ˝ is Darker and Edgier then the other. The anime version can still be seen as lighter and softer then the manga. Not only does it lack the somewhat darker storylines that the author created after the anime was cancelled... which includes, among other things, a Villain of the Week whose father was technically killed by Genma, Ranma meeting his mother — and then being threatened with Seppuku because Happosai dressed him in girl's clothes, Ranma being trapped in female form by a group of more powerful and much more vicious martial artists, and Akane being kidnapped, near-drowned and then changed into a doll, an almost fatal experience, over Jusenkyo... but also removes several of the more Comedic Sociopathy moments from shared storylines. Examples of this include Akane's viciously vindictive speech to Ranma after the first Nekoken incident, Happosai's attempt to murder what he believes is baby PantyhoseTaro because he thinks Taro will become an even better Panty Thief then himself, and the ending of the Hypnotic Mushrooms story, which in the anime ends with a gag and in the manga ends with Akane surrounding herself with weapons to use against Ranma if he "tries something" while Ranma gives her a Deadpan Snarker response.
Japanese fans of Lupin III were quite nonplussed when The Castle of Cagliostro hit theaters, as Miyazaki had made the normally-obnoxious Lupin and his cohorts unrecognizably nice. The film originally flopped at the box office. It took the passage of years, and fans who were able to see the movie on its own merits, for it to gain the popularity and critical acclaim it has today.
The same could be said of the second Lupin anime series, which is a lot goofier in tone than the original comics and 1971 anime series. Not to the degree of Cagliostro, but it definitely fits this trope.
The Getter Robo 70's anime series removes the batshit insane elements the original manga had. Ryouma becomes a virtuous, Hot-Blooded straightforward hero, while Hayato becomes the archetypical 'cool guy' loner... all in high school setting. They're still Hot-Blooded, though. But just try to compare that version to the latter closer-to-manga versions (Shin vs Neo, Armageddon, NEW)... where they're both Ax-CrazySociopathic Heroes with the side order of Hot-Blooded.
The 1972 anime adaptation of Devilman was considerably lighter and softer in comparison with the original manga: Lots of changes to the story were made, the violence and nudity were considerably toned down, comic relief was included, and the most of the character whodied at the end of the manga are still alive in the last chapter of the first anime series, which also has a Bitter Sweet Ending instead of the tragic conclusion of the manga.
Soul Eater's sidestory manga from the same mangaka Soul Eater Not! appears to be a twist of this from the first three chapters. It is focused on a period prior to Sid becoming a zombie (an event from Chapter 1 of the manga) and looks at how new normal academy students (instead of how badass seniors) fit in. Along with how students can raise money, waste money, get their names (Eternal Feather for example) and live in academy quarters.
In the Mai Hi ME manga, the only person who dies at any point is an Asshole Victim who previously tried to sexually assault Shiho. After the characters work through their competing approaches to fighting Orphans in the first arc, they work together against common enemies, instead of being manipulated into fighting each other like in the anime. The characters also suffer significantly less emotional trauma than they do in the anime. A notable example is that Akane is able to win her fight and profess her love to Kazuya without her CHILD (and Kazuya by proxy) being killed and her descending into insanity.
The Mai-Otome manga is similar, with Nina remaining emotionally stable and loyal to the heroes the entire time, largely related to Erstin surviving, and Arika and Mashiro never quite sink to the depths of despair that they do in the darker parts of the anime. On the other hand, several characters who survive the anime are killed off, and Sergay is arguably more evil than the Otome anime version of Nagi.
The Mai-Otome anime can also be considered this in comparison to the Mai-HiME anime, as while the story is, in the best of times, more serious, it never seems quite as dark or hopeless, and there are fewer character deaths, although they are more often permanent.
The anime adaptation of Rosario + Vampire definitely is this. It cuts out most of the darker story arcs that crop up later in the storyline, while considerably softening many of the earlier ones, and generally portraying a lot of antagonists as less evil than their manga counterparts. For example, the Attempted Rape element of Mizore's introductory arc was removed completely, while Ruby's master Yukata, a notable Knight of Cerebus, was reduced to a tragic Posthumous Character. Finally, Tsukune's ghoul transformation, a major contributing factor to the series' Cerebus Syndrome, was ignored completely.
The Black Jack TV anime series adapted stories from the manga, but occasionally removed some of the downer material.
In the manga story "Thieving Dog", a dog dies saving Jack and Pinoko from an earthquake, while in the anime the dog survives the incident and becomes their pet.
In the manga story "Sometimes Like Pearls", Jack's killer whale friend dies of injuries acquired while attacking fishing boats and killing fishermen. In the anime adaptation, "The Gift From a Killer Whale", the whale was hurt fighting another whale who was attacking the fishing boats, and is proven innocent just in time for Jack to save him.
Higurashi: When They Cry: Rei and Kira. Before they came along, Higurashi itself was about murder and mystery, and had various recurring characters beating other people to gory pulps. Rei becomes much lighter in tone after Takano's defeat at the hands of the club and their friends in Matsuribayashi-hen, especially with the hilarious pool episode. And then Kira comes along, and it's much lighter than Rei in every way, with things ranging from Rika and Satoko becoming Magical Girls to hilarious and absurd situations happening during dates between Keiichi and the Sonozaki twins along with the other female protagonists, and even Keiichi ending up dating Oryou, Mion and Shion's grandmother.
To say nothing of the KeiichixSatoshi yaoi joke from Kira. No, really.
The original Fruits Basket is noticeably darker than the anime. Even without trimming some drama here and there, the anime covered the Lightest and Softest part of the manga, cutting off before the main plot really even started. Besides that, the two darkest characters were also the ones that deviated the most from their characterization in the manga: Akito was simplified into a fairly standard villain, and Shigure was relegated to comic relief.
Episode 3 has Manabe beaten up significantly in the anime. In the yonkoma, he got stabbed, and was not conscious when Kotoura came to check on him. Also, Hiyori convinces the thugs to go after Manabe by telling him that he's stalking her. In the yonkoma, she tells them he raped her.
In the anime, one story arc has a criminal prowling the streets savagely beating up schoolgirls, resulting in them getting hospitalized. In the yonkoma, the criminal actually rapes and murders them.
There is a chance that the Shiki anime is this compared to the manga and, especially, the original light novel. At least one character had his worst traits from the manga sanitized in the anime to make him more sympathetic, and in the original light novel Natsuno never rises from the dead, while in the manga and anime he does. Of course, even if this is true, the anime itself is still really, really dark!
Ginga Kikoutai Majestic Prince is rather light and soft after Hirai's previous works. It has some of the most adorable funny expressions to grace a Hirai anime.