The title character of The Bastard Of Kosigan, a Burgundian during the period when which the Duchy of Burgundy was an independent state, has the option of taking offense when people refer to him as a Frenchman.
In Borderlands 2, Handsome Jack wishes to make very clear the difference between "choking" and "strangling".
Handsome Jack: No, no, Jimmy, choking is something you do when you eat too fast. As I'm crushing Mr. Moorin's windpipe with my watch chain, what I'm doing is actually referred to as strangling.
Handsome Jack: Mr. Tassiter—maybe you can settle up something for me—do you know the difference between choking and strangling?
Crysis 3: Claire insists on calling Prophet "hardware". Prophet disagrees.
Paws from Cthulhu Saves the World would like to remind you that he is not a cat, but an ultharian (a race of green feline-shaped aliens whose tails can transform into Combat Tentacles), a fact Umi won't learn any time soon.
Dalish of the Bull's Chargers insists that she's an "archer", despite it being very obvious that she's an apostate mage. She also claims that her magic staff is a "bow". The glowing crystal is for aiming; old elven trick, you wouldn't understand. The other Chargers go along with this to protect Dalish from people who don't like apostates (which is pretty much everybody).
The Qunari do this a lot whenever some of their members don't quite fit in their roles, particularly gender roles. If a woman wants to be a soldier and proves to be very good at it, she is allowed to fight, but is also considered a man while fighting. Because women under the Qun aren't soldiers. Likewise, the Qun treats mages as dangerous things (literally, the Qunari term for mages "Saarebas" means dangerous thing) that must be chained up like animals. If a mage earns the respect of the Qunari, the Qunari simply don't think of that person as a mage. They are non-mages who just happen to be able to perform magic. But they are not mages.
The Dwarf Fortress community insist on calling hell Hidden Fun Stuff. Demons are Clowns, and adamantine is cotton candy. Also, any mention of possible failure or defeat is referred to as "Fun."
In Eien no Aselia theres a kind of pastry that's basically identical to a waffle filled with some sort of sugary fruit. Lemuria constantly corrects him when he calls it a waffle instead of a yofwal, while in turn he stubbornly refuses to refer to them as anything but waffles.
In The Elder Scrolls IV: Oblivion an aspiring Orc "knight" named Mazoga insists that the player call her "Sir Mazoga" or similar, and will get angry if the player does not. Depending on dialog choices and how the player handles the associated quest, she eventually realizes she's being a pompous jerk.
In Fallout 2, the citizens of Vault City keep a slave labor force, but insist upon calling them "Servants", and make a habit of expelling those who say otherwise.
Similarly, in the Fallout 3 DLC Pack "The Pitt", The leader of the Pitt Raiders, Ashur, insists that his subordinates refer to the slave populace as "workers".
Also in 3:
Butch: I'm a barber, not a hairdresser! There's a difference!
And in Fallout: New Vegas, we have the subtle distinction between a scavenger and a prospector, along with the varying ways to say Caesar's name note Most characters say it as "See-Zer", like most Americans do. The Legion pronounce it more like the German "Kaiser" (which came from the original Caesar's name). The second one is the correct one.
Ranger Jackson is not allowed to pay mercenaries. He does sometimes accidentally supply someone who isn't technically part of the NCR when they have been helpful enough for him to forget their status.
Mr. House doesn't hide the fact that he'd be the sole ruler of an independent Vegas, but he'd be an autocrat, not a dictator.
As far as Raul is concerned, the city that people now refer to as Two-Sun is pronounced Tucson.
Cid in Final Fantasy II wants to make it perfectly clear that he's lending you his airship. But seeing as those are his last words...
Locke of Final Fantasy VI. He's not a thief, he's a treasure hunter. And he'll rip your lungs out for saying otherwise! What makes this one line particularly funny is that the SNES translation is so aggressively kid-friendly that it refuses to acknowledge the existence of pubs and the word "death" and its variants... but threatening to pull out somebody's internal organs is A-OK?
He's much less violent in the GBA remake. He'll just beat the crap out of you.
The Legend of Zelda: Phantom Hourglass. It is implied that Tetra got a lot of those after it was revealed that her real name is Princess Zelda, but in the game itself it only happens in one short exchange, where she told Niko to keep calling her "Tetra" instead of the other name. Tetra's level of anger about this seems to imply that it has happened before and started to annoy her.
Tali: (completely plastered) Veeeeeeery carefully. Turian brandy, triple filtered, then introduced into the suit via an emergency induction port.
Shepard: ...that's a straw, Tali.
Tali: Emerrrrrgency. Induction. Port.
In Mass Effect 2, it is revealed that all quarians refer to the leader of a ship as "captain", regardless of their actual rank: while Shepard never made the rank of captain, they are still responsible for the lives of the crew of the Normandy, hence, Captain Shepard.
The Matrix: Path of Neo has an Agent who insists on calling an ex-cop turned security guard "Officer" because he's trying to get him to kill Neo.
Agent: Do your duty...Officer, kill him, kill him now.
Another example with a girl who insists on calling Neo's cassock a "dress".
Mortal Kombat: Sub-Zero would like you to know that he is a Lin Kuei assassin, not a ninja.
Phoenix Wright is a lawyer. Not a dentist, policemen, detective or parliamentarian. And no, his badge is not made of plastic and not available in a Gumball Machine. (He has to insist on those facts so frequently, it actually strikes him as strange whenever anyone actually gets it right.)
Yet when you fight him, he's still called "Leader" Wake, just like any other Gym Leader. This is mostly due to character restrictions for in-battle text.
In the Professor Layton games, don't bother trying to call Luke Triton anything other than Layton's apprentice. He simply will not accept any other name for their relationship. He even cuts off Layton himself whenever the professor tries to clarify it.
And in Professor Layton and the Curious Village, Layton repeatedly corrects that he is not, in fact, a detective. This doesn't stop the people of St. Mystere from constantly thinking that's the case, though.
In Rune Factory Tides Of Destiny during Joe's second frienship event Sonja calls Joe a 'pervert' for looking through a crack in the womens' bath. He insists that he is a 'treasure hunter' who is after 'a different sort of treasure'.
The students and professionals in Senran Kagura are Shinobi. The promo materials occasionally call them ninja, but never kunoichi. While the latter is a decently well-known term for female ninja even in the west, it carries the connotation of relying on deception and feminine wiles, which Shinobi just don't do.
In Sid Meier’s Alpha Centauri, the humans that have settled the new planet have vowed not to repeat the mistakes that turned Earth into such a Crapsack World, and as such they will not declare war on one another. Factions will have prolonged periods of armed conflict against each other when they become displeased, but they call it 'Vendetta' instead.
Meta-example for Star Trek Online. Cryptic insists the Klingon Bortasqu'-class is a "battle cruiser" despite the fact that stats-wise it's really a Starfleet-style cruiser with the serial numbers filed off: an engineering/tactical Mighty Glacier. This led to them not giving it all four cruiser commands when they debuted (although in practice "Attract Fire" isn't that useful in the PVE metagame anyway).
In Super Robot Wars Original Generation, Dr. Marion Radom flat out refuses to call Kyosuke's and Excellen's machines the Alteisen and Weissritter, instead using their original, production-model names — the Gespenst Mk. III and Gespenst Mk. II Custom, respectively.
Until OG2, anyway, when she's impressed enough by their performance to call them by their codenames. Her selective hearing still keeps Excellen from getting a proper upgrade for "Weissy," though.
"It's Magnius from the eastern ranch!" "That's LORD Magnius, vermin!"
And of course, the sequel has the Nazdrovie / Light-Frog debate between Centurion Tenebrae and the main cast. The main cast start out calling it the latter while Tenebrae insists on the former. Being that Tenebrae is normally The Gadfly, they decide to get back at him by purposefully using 'light frog' until he gives up and starts calling it that too, at which point they suddenly start calling it a Nazdrovie instead.
Team Fortress 2. In "Meet the Sniper" video, the Sniper is discussing his career choice with his unseen father over the phone, and it's evident from the Sniper's exasperation that they've had this conversation many, many times before:
Sniper: Dad? Dad, I'm a—Ye—Not a "crazed gunman", dad, I'm an assassin!... Well, the difference bein' one is a job and the other's mental sickness!
With Halloween 2011, we now have MONOCULUS!, all caps and exclamation point necesssary. Upon being killed by him, players must raise their hands to the skies and shout his name (then explain to the cops when they show up).
Lara Croft of Tomb Raider uses a more honest euphemism than most examples of this trope. "Tomb Raider" can be converted to "Grave Robber" just by replacing words with their synonyms.
Marisa Kirisame of the Touhou series doesn't steal things. She borrows them. Without permission. For the rest of her life.
Dr. Hello in Treasure Hunter G gets rather peeved when Red addresses him as just that, insisting on being called "the last remaining mad scientist, Dr. Hello"
Zorbak is an "ebil" moglin Necromancer present in several original Artix Entertainment games, and he frequently corrects people who call him evil. What, exactly, seperates evil from "ebil" is up for interpretation, but, if the little blue troublemaker's actions are anything to go by, being "ebil" seems to mean being more a Stupid EvilJerk Ass than a real villain in one's own right.