Unless the fiction already had me/us looking for something in particular, my answer to this would nearly always be "Nothing in particular, just seeing if there's anything that might be of interest".
And by that what I'm really saying as a player is "On a closer look is there anything noteworthy here that your initial narration didn't hit, or that was forgotten*, or that isn't obvious on a quick glance but becomes so on a longer look?"
* - and yes, forgetting to narrate something important happens to all of us now and then - don't deny it!
This is consistent with what I posted earlier: those "open-ended" knowledge/perception checks are really request for more prompting/content-injection from the GM. That's why I'm generally not a big fan.
In my games I'll usually give people options, some potentially gated behind an appropriate check. I don't want to get into a guessing game with the player trying to find the right thing to look for. The player can also follow up with other things.
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I want to be flexible but I don't even try to explain every single detail, I don't know how you could. At the same time the player can't create something I don't think should be there. For me the key is that my world feels real and consistent
To me at least, this suggests an approach similar to Lanefan's, where the GM's conception of the fiction is a focus-point for player action declarations, at least some of which have the purpose of eliciting more of that conception from the GM.
On the whole that is an approach that I try to avoid.
I think there is an inescapable place for fictional positioning as well, since clearly these are games where some degree of cleverness seems to matter. I mean, its theoretically possible for the fiction to be nothing more than a fig leaf, but I doubt even you will find that terribly satisfying in the long run, nor that you really play that way.
In some of the systems I play (eg Burning Wheel, Prince Valiant, to an extent Classic Traveller) fictional positioning factors in as a bonus die (or perhaps bonus dice) on a check. It doesn't factor into the "say 'yes'" decision, which is based on narrative considerations.
In MHRP/Cortex+ Heroic, fictional positioning only matters to resolution if it has been "mechanised" in some fashion, whether as a Scene Distinction or by someone creating an Asset. This can have the effect of turning some of the fiction into nothing more than a fig leaf. It certainly dials down the grittiness!
In a 4e skill challenge, fictional positioning can effect difficulties, but it's bigger influence is on how checks are framed and consequences narrated. This is a little bit like its role in AW or DW.
In any system genuinely asinine and blind-to-game-world declarations are trivially "say 'no', and don't roll". The line becomes blurrier and broader in systems designed for generic rather than specific narratives.
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Having by whatever means reached "say 'yes', or roll" a DM can then again consider advantageous positioning in the following ways. Positioning can raise or lower stakes, pointing to "yes". Positioning can be an input to parameterising "roll", pointing to better or worse odds.
Your first category of action declarations -
genuine asinine and blind-to-game-world - typically are not a big issue for me. I see that more as something that comes up in club-style games with players who don't take the game or the medium very seriously.
If a declaration is based on genuine confusion, among sincerely engaged participants, as to what the fiction of the game is, then that can be worked out via conversation. The GM's voice will obviously be important in that discussion, maybe even the most important contribution, but I don't think a conversation like that is going to be resolved simply by a GM's exercise of authority.
Your second paragraph raises different sorts of possibilities that seem apposite in various different RPGs. Although the stakes you have in mind look like they might be "procedural" rather than "dramatic"/"narrative" stakes - eg like
the chance of falling that
@Lanefan has posited as inherent stakes in climbing a wall. My preferred approach to "say 'yes' or roll the dice" is the BW approach (itself derived from DitV), where the focus is on narrative stakes and not procedural ones.