The word "precisely" is yours, not mine.
But anyway, the players aren't "interacting with a fixed world". That's metaphor. What they're actually doing is prompting the GM to say things. The GM is, obviously, the author of what they say. So if the players are controlling it, they must in some fashion be constraining or guiding what the GM says.
One way to do that, that I know of, is to have determined priorities for their PCs that the GM then responds to.
Another way is to have sufficient knowledge of
what it is that the GM will refer to when deciding what to say that they can trade on that knowledge to prompt particular responses.
Maybe there are further ways, but I don't recall them having been suggested in this thread.
Consider the simple example that
@TwoSix posted upthread: "You start in a room. There are doors to the north, east, and west. What do you do?" In this situation, the players know that, depending on which door they open, they can prompt the GM to say some or other thing. But they don't know what that will be; so opening a door would not involve the player exercising any control over the shared fiction.
This is why Gygax emphasises information and planning. If the players listen at a door, or use an ESP ability, or similar, then they can prompt the GM to tell them things about what is on the other side of the door.
Now they are in a position to exercise control, because they can exploit that knowledge to control or guide what the GM says.