"... as you imagine you or that other person would... " - seriously? You're not playing yourself in the game (with rare exceptions where a game specifically expects you to play as real-world you), you're playing "that other person".
I was giving a general definition of role playing, not necessarily specific to D&D, or even tabletop role playing games.
And so, a further clause needs to be added to your definition of role-playing: "...given such knowledge and information as that person would have available."
I disagree. Many forms of role playing have no such requirement, and in fact there are forms of role playing (for instance, role playing as part of therapy, or as an educational tool) where attempting to isolate the knowledge of the person playing the role from the knowledge of the character would be counter-productive.
Perhaps, but I posit that said role-playing has by that point lost some or all of its integrity.
What inherent value does the “integrity” of role playing have? Or, if you prefer, what ill comes from violating that integrity?
Most (or nearly all?) of the time, using knowledge of 'extra' information known only to the player but not the PC gives the PC an in-fiction advantage it wouldn't otherwise have. To me this trends toward cheating.
It’s only cheating if it is in violation of the rules. Which it may well be in your D&D game. It is in many people’s D&D games. But I’ve found that by permitting it in my D&D games, the experience has only improved. Your mileage, as always, may vary.
By the same token, it's also very much the case that DMs have to carefully self-police in how they run their NPCs; as the DM always knows tons of stuff the NPC would not.
Again, what harm comes from a DM not doing so?
I agree; in fact that was kind of my point. I was responding to a post suggesting the reason for any check be told to the players before the check was called for, and gave an example of how this would quicky fall apart.
Better yet, ask for the stealth check anyway whether there's observers there or not...in this case, as the character is trying to move through a crowd, the check might also inform whether the PC somehow caught the attention of a random passer-by - e.g. the PC is trying her best to be stealthy and some little kid yells out "Hey - why is that person acting so sneaky?!"
It breaks character the moment they do something they wouldn't have done otherwise, or change what they'd already committed to doing.
Example: party says they're going to sneak down a passage past several open doors. Not until the 4th door do you call for a check (you-as-DM already knew the first three held no threat); and in response, before rolling, someone says "We stop here and rearrange our marching order into battle formation".
Now you-as-DM have given yourself a headache. Do you ban them from changing their order based on their prior commitment to sneaking the length of the hall? Do you let the order change happen and thus set a precedent that such metagaming is allowed?
See, I as a DM wouldn’t do that. I don’t call for checks except in response to actions described by the player. So, I wouldn’t just say, “make a Stealth check” when the player passed the third door. I would give them that Metal Gear exclamation point first. Maybe I describe a “what was that?” Coming from the other side of the door, and then ask the player what they do. That insures I’m asking them what they do
about something, and the check, if needed, resolves what they do about it.
Wouldn't it have been better to call for the stealth check at the first door, even though there was no threat there? The characters (in theory) wouldn't know which doors held threats and which did not, so why not determine their SOP at the first opportunity?
They can still interact with the game, only that interaction is going to be based on less-than-perfect knowledge - and this is quite realistic, in that their PCs wouldn't have perfect knowledge either. There's always something to interact with, only sometimes that thing is just a shadow.
It may be realistic, but realism isn’t my goal. My goal is to create opportunities for the players to imagine themselves as other people, in a fictional scenario, and make decisions as they imagine those people would do in that scenario. Telling them to make Stealth checks as they pass doors that enemies may or may not be hiding behind to preserve a sense of role playing integrity does not, in my opinion, serve that goal. Telegraphing that they might be in danger of being detected and asking them what their character does about that serves that goal very well (again, in my opinion.)