While I agree that people will approach how they play an RPG or how they "inhabit" their character differently, I do think it's important to clarify something regarding your categories.
Roleplaying is literally playing a role.
That we agree on.
It doesn't require acting, which is a different skill that can be applied.
That we agree on, too.
It's simply making decisions and choosing actions as if you were that person.
Here I disagree somewhat. Or, at least, I disagree with what others seem to mean when they say similar things. I'll elaborate below.
This is the same if it's a roleplaying exercise in a training scenario at work as it is an RPG. Category A is definitely roleplaying.
So Category B is only roleplaying if the role you are taking on is you. There's nothing wrong with playing a PC as if it is you within the presented scenario.
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In terms of the fire and trolls, my only comment is that most of the time, those that argue that a specific PC won't know that trolls are vulnerable to fire is not considering the fact that within that setting it's likely that civilized people have been fighting trolls for thousands of years and it's not likely to be a secret. While a player might occasionally me ask whether they would know something or not, the only time that we ever comment that somebody won't know something is when the party is split up (it happens often), and a player (usually accidentally) acts based on info from the other group. It's very rare any of us call anything out because we feel the best person to determine whether their PC knows that "trolls are vulnerable to fire or not" or similar scenarios is the player themselves.
If the player honestly believes that they would not know to use fire on trolls, and thus they choose not to, they are roleplaying their PC just fine.
The second situation, "...for his sake I'll pretend..." is different. It's certainly not wrong from a gameplay standpoint if you are intending to benefit your fellow player. But it's no longer roleplaying by definition, because you aren't making the decision or acting based on that character that lives in that world. Instead, you are choosing a course of action based on what's going on between players at the table.
That doesn't make it bad, so don't take it as a criticism. But if the goal is roleplaying, then really the only question you need to answer is, "what would this character do?"
My general recommendation, which is largely lost in the current D&D (and other) rulesets, is to help players to better understand their characters as real people in a real world. The more you understand your character, the better you can roleplay that character. And within the context of the game, nearly everything that isn't directly interacting with the rules, can (and should?) be roleplaying. Combat is still roleplaying. When you engage in combat, how you engage in combat, the actions you take, everything is still roleplaying. The only part that isn't roleplaying is the mechanical resolution. How you react to that resolution is.
Most of this I agree with. The point of departure is in determining "what would this character do". Because there is no one thing. Minds are not that deterministic.
You say, "the player honestly believes that they would not know to use fire..." I mean, sure, the player could
believe that. But I prefer to ask, "What's the coolest thing this character
might do, and is there a reasonable justification for it?" You can
decide what your character believes, there is no right answer. And even if you could know the most
likely thing your character would believe, wouldn't that also likely be the least interesting choice?
I like to use the example of Bilbo going on an adventure. That was actually not what a Hobbit is most likely to do. A Hobbit is most likely to refuse to go on an adventure. (And the rest of the story was full of things that Hobbits are unlikely to do.). I'm really glad Tolkien did not restrict himself to asking "what is the most likely thing a respectable, middle-aged Hobbit would do in this instance?" because it would have been a short, boring story. Sure, oftentimes, especially early in the story, Bilbo did do the most expected thing. But that changed throughout the story as he increasingly did more and more surprising things. Especially when it mattered. Bilbo continued to act like a respectable Hobbit, in order to portray his Hobbitishness, but primarily on inconsequential things
The response I often get is, "No, going on the adventure was actually the most
likely thing to do, because Bilbo isn't just any Hobbit, he..." and then they list the reasons Bilbo is a special Hobbit.
Exactly. And your character is not just any commoner, he/she is a protagonist of the story.
So, again, while I acknowledge that it is a form of roleplaying to decide what your character "would" do, I think that "would" is entirely subjective.
But it is just as much roleplaying to ask "What
might this character do, and how would their personality explain that choice?" And, "How does taking this unexpected course of action help me to flesh out this character, and make him/her more real?" And even, "How does this choice suggest that my character is evolving into something different than he/she was earlier in the story?"
And I find that sort of roleplaying to be much more interesting, and to result in much more engaging stories, than to simply ask, "What would he/she do?"
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