RPG:
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Role-playing-game
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Game in which you play a role
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Practically every game in existence
And this is the mistake everyone makes in my opinion.
Lets go back to Doom and Bioshock: Infinite and the idea of "assuming a role." These are game pawns you control, nothing more. They are the shoe in Monopoly. At best you are joint role-playing these characters along with the script writer and voice actors. Even when playing with a premade PnP character at a convention, I get to choose how to express their personality and mannerisms, their voice inflection, I get as many ways of handling a situation as I can come up with within the limitations of my avatar, etc. Even when that premade character's personality is defined for me, I still get to be the writer and performer. The closest video games have come to this is allowing you to choose from different dialogue choices, choose a win pose, shit like that. Doom and BI offer none of this. They are weapons in space, tank turrets you control.
But if we're just going to list features, I think observable data about your character is important to i.e. the character sheet and stats. Your stats are an assemblage of your intrinsic, naked attributes -mental, physical, social, etc; the composition of your character before any gear is applied to them. The numbers represent the value of these attributes on the world scale, so if say 20 ST is the maximum strength any living being can have in the game world, and you have a strength of 18, you are pretty fucking strong (just an example). In a RPG, you have complete knowledge of all your capabilities within the world scale. But how do we know just how strong or intelligent Doom guy is, or Booker DeWiit? The feedback for this is very limited in the game. You can't know exactly what you are, so your performance is limited to mere mechanics of movement and attack.
The intrinsic attributes part is also important in sorting through the confusion. Many people have argued that Legend of Zelda is a RPG because you have hitpoints in the form of hearts, and you get stronger by getting more hearts. But how you get more hearts has nothing to do with how strength/durability is gained in the real world, which RPGs try to emulate (within the limits of not having your character die in one round form a axe strike, which would make for a less satisfying game). In the real world, you improve your bodily and mental attributes by taxing them so that they adapt and grow stronger. In DnD, you do this by fighting and your defensive attributes go up. The heart containers in LoZ are essentially power ups, not Link's body's response to stimulus. Of course DnD and others aren't perfect or even consistent, like why should you be able to raise Charisma if you leveled off of combat? Still, your character gets better through resistance, not pick ups.
In my view, RPGs in the context of videos games are those that emulate, mechanically AND cosmetically, their PnP grandfathers. So even though you don't have character performance in Wizardry, mechanically and cosmetically the game fits squarely into the RPG genre and nothing else. Character sheet, party, stats, random die rolls, to-hit chances, turns, iniative, etc. And there is observable information by which you can gauge a lot of this. Not so in Bioshock, Saint's Row, CoD, Uncharted, Ninja Gaiden, Mario Bros. Pac-Man, Pitfall, etc.
We can go into permanency of attributes and skills and things like that, too. Someone may argue that CoD4 has a character sheet with your inventory, perks, etc. But perks are just more power ups. They don't represent any innate characteristic of the character, you can swap them repeatedly. There is no attempt at simulation of how people learn things, increase and maintain skill level. And you still don't have information regarding just where your mind and body stand in relation to everyone else, nor are there any requirements to know these things. Traditional JRPGs, being limited, only pay lip service to a lot of this information, but cosmetically it's there. There is no confusion as to what genre they are trying to fit into.
Perhaps most importantly, Gary Gygax was terrible at naming shit.