As I said, I'm not that interested in quibbling over terminology.If it's the player's idea, it's player-driven. It might still be GM-approved.
But when I play a RPG, I prefer to be the one driving what is possible for my PC.
And as a GM, I prefer the players to be the ones driving what is possible for their PCs.
I'm not really the right person to ask, as it's not a distinction I'm very invested in. I think a lot of people call the first thing "linear" and the second "sandbox". I don't find that terminology very illuminating - I think nearly all RPGing is linear in the sense that it consists of a time-sequence of events (both in the fiction and at the table). But some others clearly do.What terminology would you use to distinguish between GM-driven games where the content is (to use your phrasing from an earlier post) "a story that [the GM] wants to tell" versus GM-driven games where the GM does not have a story in mind and instead "decides and curates content" in response to the players' action declarations for their characters?
Although I agree that both of these styles have more in common with each other than they do with styles where the players have explicit narrative control, I don't think they're similar enough to lump together into the same bucket except at the most general levels of comparison.
What I would find helpful is for advocates and practitioners of sandboxing/"living worlds" to sometimes write a bit more plainly about their techniques. For instance, talking about I look for my brother or A lengthy investigation are both describing in-fiction events. But what does these look like as action declarations at the table? The more the answer is just questions posed to the GM, which the GM answers by consulting his/her notes and/or extrapolating from those as seems logical and fun, then - in my mind, at least - the less the difference between the two approaches that you are wanting to distinguish.