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.
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.
Jade: Why if it isn't Dist the Runny! Dist: The Rose! R-O-S-E, rose! Dist the Rose! Anise: You mean, Dist the Reaper. Dist: Silence! I refuse to accept that name! It's Rose! ROSE!
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 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.
An example from Gabriel Knight 2: "I do not know what it's like where you're from, Mr. Knight, but people refer to me as Herr Doktor Klingmann here."
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.
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.
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.
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"
Tear from Recettear is a loan shark (not quite that bad, she's nice and helpful about it) but doesn't like to be called as such.
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.
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.)
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).
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.
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.
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'.
Fate/EXTRA gives you the option to beat on of the most powerful characters in the Nasuverse, they balance this out by telling you at every opportunity that the only reason you can win is because her Master is a raging moron.
In Katawa Shoujo, this comes up in a context surprisingly unrelated to any disabilities. Lilly and Shizune both insist that their Big Fancy Houses are not "mansion(s)".
Lilly Satou, after Hisao realizes she is a Covert Pervert, will say that she has "a healthy adolescent sex drive".
This pops up in an amusing conversation between Shepard and Tali in Mass Effect 3:
Jak II: Krew prefers the term "freedom challenged" over "slave".
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.