Going to address this out of order to better prioritize my thoughts on the matter:
Yep. Which is kind of the point - is the player using player info or character info and can (and will) he keep them separate.
If the player info and character info are kept truly separate, then knowing what the player knows is never relevant. All that is relevant is whether the action the character is taking is one that the character could choose to do with the knowledge that the character has.
If a player can't have their character do [insert in-character action] and the reason why they can't do that is because the player knows [insert piece of information], that is forcing the character's actions to be determined by the player's knowledge -
not separating the two.
So no, policing the player's thoughts isn't "kind of the point" according to any definition of metagaming that I've ever seen anyone use - it's the unintended and counter-productive result.
Which, if we are to follow your line of thinking, means we need some new terms for specific elements of what used to fall under metagaming.
There are actually already terms for the things which fall under the outdated definition of metagaming.
[term 1] = when a character takes actions based on knowledge it could not have (e.g. scout goes out, DM tells table what is found, but scout dies before reporting in; characters act as if scout reported in anyway. The troll-vs.-fire example fits in here also)
Firstly, the troll and fire example has nothing to do with this: one does not need to know anything other than that fire is available, fire is a dangerous and destructive thing, and they are under attack in order to choose to use fire as a weapon against their attacker.
Secondly, a character performing actions that it absolutely could not possibly have a reason for doing is called "cheating" or "not playing in good faith."
[term 2] = when a player takes actions based on knowledge s/he should not have (e.g. overhears DM telling a friend some plot info, then uses that info in game. Or the player knows what a monster is but the character does not, the troll example again)
First, is not relevant in my troll example whether the player knows their character faces a troll, and that trolls are especially susceptible to fire. The character could attack with a burning log if the monster were an ogre, and especially large gnome, a cleverly disguised fire elemental, or even the avatar of the principal of logic.
The majority of these actions that get called "metagaming" are actually one of two things: A) completely acceptable play that has triggered a false positive in someone's metagaming sensor (this sort being the things that an experienced and knowledgeable player gets harangued for while a completely unknowledgeable and inexperienced player doing the same would not cause a response), or B) cheating or not playing the game in good faith.
[term 3] = when a character takes action based on mechanical dice odds rather than the in-game situation
If you mean making a choice to use the greatsword a few feet away from the character because it does 2d6 damage instead of using the flaming log in the character's hand because that would only do improvised weapon damage plus 1 point of fire, then you have hit the nail on a thing which is actually metagaming. The character would regard both a greatsword and a flaming log as dangerous and potentially lethal weapons, because which one does "more damage" wouldn't actually be clear beyond "a bad run-in with either will kill you."
Or when a player of mine insisted that he couldn't possibly escape with the rest of his party because he's a dwarf, so he "had to" turn and fight the overwhelming odds that the party had initially planned to flee from. He was thinking of the movement speed listed on his character sheet, not the in-world reality that sometimes people do actually retreat and get away even if they are dwarves.
I don't know what else you could have been meaning by that, so let me know if my examples aren't the kind of thing you meant.