Attack on Titan has Bertolt drawing a line between "soldiers" and "warriors", especially when discussing his childhood friend, Reiner. It's one of the hints that both are The Mole, drawing a line between the human "soldiers" they've infiltrated and their own group, identified only as "warriors". These terms help to keep track of Reiner's mentalstate as he goes insane from guilt.
Kotomi from CLANNAD insists on being addressed with "Kotomi-chan" and won't even register other forms of address. This makes most people somewhat uncomfortable, as it is a rather intimate form of address.
In the same show, Fuko's carvings are starfish, not stars.
A very poignant use crops up a lot in Code Geass; the racist, darwinist Britannians call those who were once Japanese "Elevens", since Japan was redesignated "Area 11" after the Britannians invaded, as a way of oppressing the Japanese; They go as far as to illegalise the use of the terms "Japan" and "Japanese". It is frequently done by two characters:
In Gintama, Katsura constantly corrects Gintoki, who calls him "Zura", to the point where it's his Catchphrase. The joke evolves over the series into silly puns, so when someone calls him anything other than his real name, Katsura says "Not X, it's Katsura!" He sometimes changes it to "Not Zura, it's X" if he's wearing a disguise, though he blows his cover by saying "Not X, it's Katsura!" just as often.
GUN×SWORD: "Carul-san..." "Carmen, Carmen, Carmen 99!"
Assistant chief security maid Yashima Sanae in Hanaukyō Maid Tai: La Vérité. In Japan "Sanae" is normally a first name, so people often call her Sanae instead of Yashima. She always corrects them when they do so, telling them that Yashima is her first name and Sanae is her last name.
Of course, the protagonist of Kenichi: The Mightiest Disciple would never run away from a fight. He may occasionally make a strategic withdrawal in order to stay alive, but run away? Never!
In Magical Girl Lyrical Nanoha A's, Zafira insists that Arf refer to him as a "guardian beast" rather than as a familiar; Arf argues "It's the same damn thing!" Given that the Belkan magic system's terminology differs from the more common Mid-Childa system in many ways ("knights" instead of "mages"; "knight armor" instead of "barrier jackets"), Arf is probably correct that there's no actual difference.
Albireo Imma of Mahou Sensei Negima! likes his tournament alias of "Ku:Nel Sanders" so much that he requests everyone to call him by that name and will pretend to ignore you if you call him by his original name.
In Haruhi Suzumiya, the SOS Brigade is not a club, as that requires approval they couldn't get.
A more meta example: The cast of My Monster Girls Too Cool For You includes several youkai. While they are more properly called yukionna, nekomata and ningyu in Japanese mythology, the author insists in calling them "the snow youkai," "the cat youkai" and "the fish youkai," so as to avoid offending the actual youkai in question.
Sasuke corrects Deidara that his jutsu is called "Chidori", not "Raikiri" (Lightning Blade). "Chidori" is the formal jutsu name, and "Raikiri" is the nickname given to Kakashi's version.
On a number of occasions, Shikamaru and Shikaku corrected their opponents by insisting that their technique is called Shadow Imitation Technique (Kagemane no Jutsu) and not Shadow Bind Technique (Kage Shibari no Jutsu). The Shadow Imitation Technique is more refined version of the older Shadow Bind.
In a flashback on the day of the Uchiha massacre, Sasuke's mother offered to help him with "shuriken practice" after school. Sasuke then insists, "It's not practice, it's training!" His choice of words is implied to reflect the desire that his efforts as a ninja in training be taken more seriously, since, when he was young, he felt as though he was living in the shadow of his brother.
One Piece: Sadi-chan insists on being referred to as such ("Miss Sadie" in the Viz manga)
In the Punk Hazard arc, the corrupt Vice-Admiral Vergo demands that Law call him "Vergo-san," a form of respect due to their shared past and Vergo's higher position in Doflamingo's empire, and beats the crap out of him whenever he doesn't. Ironically, the one time Law does call him "Vergo-san" is when he regains his heart, and before proceeding to cut him in half.
Pokémon: Butch from the anime series constantly corrects those who get his name wrong. Everyone has mistaken his name for either "Biff", "Bill", "Bob", "Hutch", "Butcher", "Patch", "Botch", or "Chuck". Whenever his partner, Cassidy, gets it right, he accidentally corrects her with the wrong name.
They also get orders from Dr. Nanba, who has the exact same problem as Butch.
As does Stephen (not Stef-AN or Steven but STEPH-an) in the Best Wishes saga.
Brandon demands that he be called "Brandon", not "Mister".
In Queen of the Serpentine, one of Lucy's apprentices insists that Lucy be addressed as "Pike Queen Lucy".
Lina Inverse insists on referring to Philionel El Di Saillune as the "First Royal Successor". Don't call him "Prince" in front of her. Just don't. In the novels, this is partly because, thanks to some succession issues with Saillune's royal family, this is his actual title. He really isn't a prince, even if he is the current ruler's son.
Not to mention the number of people Lina has to, ahem, correct about the various titles she's usually introduced with. Generally by the assembly of local banditry.
Steins;Gate: The name of Future Gadget #8 is "PhoneWave (name subject to change)", because Okabe really dislikes the name "PhoneWave" and refuses to acknowledge it as permanent, but at least for the moment doesn't have a better idea. He will correct anyone who leaves the "name subject to change" off.
Same thing in Tenchi Muyo!: "No! I'm not gonna help you unless you call me Little Washu!" ("Washu-chan" in the Japanese version).
Greatly exaggerated in the minds of fans. Washu did this once in Tenchi Muyo! Ryo-Ohki, and six episodes later it came back to bite her on the butt when she discovered that Mihoshi had included it in her report. That hasn't stopped fanfic writers using it to death.
Manjoume has to constantly correct anyone who doesn't use honorifics when saying his name with "Manjoume-san da!" (translation: "That's Mr. Manjoume to you!"). Unfortunately for him, everyone mistakes this as "Manjoume sandaa", instead ("sandaa" being the Japanese pronunciation to the English world "thunder"). While he hasn't stopped with the correcting, "Manjoume Thunder" has embraced the mistake as part of his personal motto. In the dub, Manjoume's counterpart Chazz Princeton does the same thing, insisting everyone call him "TheChazz".
Dr. Chronos was like this too. In the first season, he insisted on being called Doctor Chronos to anyone who addressed him otherwise. In the second season, he became Chancellor of the school, and while students managed to address him as "doctor" now, he insisted on being called by his new title. (By the fourth season, when he became Vice Chancellor, the students had much more respect for him, and this wasn't an issue anymore.)
This happened twice in the dub during Judai's duel with Princess Rose; she was annoyed when Kenzan referred to her Des Frogs as reptiles (frogs are amphibians) and again when he called her D.3.S. Frog a toad. (Frogs are not toads.)
Yu-Gi-Oh! ARC-V: Since the Japanese name for "Fusion" is "Yugou", a lot of people either mistake Yugo's name as "Yugou" or he thinks that they mispronounce their name even if they don't mean his name, he always says "Yugou ja ne, Yugo da! (It's not Fusion, it's Yugo!)". Yugo also has a habit of correcting other things, e.g. "It's not a (motor)bike, it's a D-Wheel", "It's not 'Kendama', it's 'Maken Dharma'!", etc.
Suzuka from YuYu Hakusho somewhat fits this trope during his first appearance in the Dark Tournament. He insists on being called "The Beautiful Suzuka" and promises that anyone who doesn't refer to him as such will not live to repeat their mistake. When the foxgirl announcer Koto just calls him Suzuka, he shows his annoyance by hurling a razor sharp playing card at her head. She manages to duck as the card skewers the demon sitting behind her, scaring her enough to use Suzuka's title and constantly compliment him and his techniques during the match. And even once a whole two sagas later.