A DM can decide that the task or the intent is impossible. "I shoot the moon with an arrow to intimidate the elven archer". Nope, you can't shoot the moon but there might be another way to intimidate the archer. Task auto-fail. "I persuade the king to give me his crown". Nope, the king is not giving up his crown under any circumstances (including if you roll unbidden and get a 20, but that's another thread altogether), however you might be able to persuade the king to do something else. Intent auto-fail.
Out of curiosity, would the "Say yes or roll the dice" resolution style allow a PC to attempt something typically impossible in a game world, like shooting the moon with an arrow?
No, my understanding of systems that use that principle, like Burning Wheel in which it is phrased as "Roll the dice or say, 'yes'", for an action declaration to be permissible, it needs to pass the following two tests, both of which are decided (informally) by table consensus: a) it has to be within genre, and b) it has to have whatever fictional positioning is required. Depending on the game, your "shooting the moon" example might fail the first test, unless it's a mythical type setting where characters do things like that, and would probably fail the second too, unless the character has an ultra long range moon-shooting arrow or something. Notice I said "table consensus". This is not a power the GM can exercise unilaterally.
It would seem that hitting the tree with a rock certainly was the goal resolved by the dice roll. We're just layering on the distraction of the guard as a secondary goal that may or may not be satisfied by hitting the tree with the rock. But that seemingly is at the whim of the GM in this scenario. Perhaps we're just making distinctions here that may not be so different in the end.
I think it's important to keep in mind what the player's action declaration was. The player described their character aiming and throwing a rock at a tree. That's the task. Why are they trying to hit the tree with the rock? Is their intent just a matter of wanting to hit a tree with a rock? I don't think so. I think it's clear the intent of throwing the rock is to distract the guard. Task resolution leaves the achievement of that intent at the whim of the GM. Conflict resolution, on the other hand, maintains the relationship between task and intent. If the character hits the tree, they distract the guard.
I guess... I would find that play experience tedious.
Me too but perhaps for different reasons. I like to play by declaring actions for my character based on a common understanding of the fiction. If the GM has decided my actions aren't going to work based on information they haven't given me and that I have to poke around trying to find out, I'm going to get a little annoyed.
I'm not sure what the "but" here is distinguising. If Conflict Resolution is not "calling for rolls when the outcome is uncertain", what is it? Honest question.
Conflict resolution is maintaining the relationship between task and intent. The DM, in 5E, is free to break that connection by unilaterally deciding the task has no chance to accomplish its intent, often by referencing information hidden from the players. That's why the ability check system is task resolution unless you import something like "say yes or roll the dice" from outside the game.