As a general question, do people discussing the "I get to decide that my character knows x fact" issue distinguish based on player knowledge? As far as I can see, it would be just as legitimate to decide that your character just knows how to stop a troll from regenerating, or which statue the treasure is hidden behind even if you as a player don't.
Great examples.
But you're conflating
thinking you know something with
knowing you know something.
So, for example, let's say you've read the published adventure and you "know" where a secret door is. If you use that knowledge, and the DM says, "Nope, sorry, you don't discover anything?" are you going to feel cheated? Or are you going to shrug and assume the DM has altered things? (I'm hoping your answer is B. If not then I'm not sure we are ever going to make progress here.)
You only
thought you knew where a secret door was, but you were wrong. Note that nobody is preventing you from taking an action with your character. The only thing the DM adjudicates is the results.
Now back to your examples. If you're in my game and announce, "My character knows how to stop trolls from regenerating because when he was a wee lad he was friends with a mad scientists who told him the secret" (or whatever) I'm going to say, "Great. What do you do?" At that point you are free to take any action you want (and if it's really good I might even have it stop the trolls from regenerating). If it doesn't work, you are free to conclude the mad scientists lied to you, or whatever you like.
Same thing with the statue: go ahead and narrate all you want about how you "know" where the treasure is. The rubber meets the road when you pick a statue and look behind it.
But in all those cases I, the DM, am not dictating to you what actions you may take, or what thought process goes on in your character's head. That's all up to you. I simply control the effects your actions have on the game world.
And where does player agency in deciding what their character knows fall on that line?
Again, there is a difference between what you
think you know, which is entirely up to you, and whether what you think you know is actually true in the game world.