I'm trying to be 100% constructive here - which is hard to do while disagreeing.
I'd suggest that the D&D play being described transcends the description of choosing from multiple options. Some options may very well be presented (often for practical reasons - as some players just don't have the 'it' whatever the 'it' is to come up with something they want to do in the game completely on their own - and I know because in the past I've often struggled with that..) But as long as you are in a game with players that can come up with what to do all on their own - then a menu of options aren't even necessary. I don't know of a better term for this kind of play than 'player driven'. In many ways it's very similar to what you call 'player driven' play.
That said I think this kind of 'player driven' play is also very different in some respects from what you call 'player driven'. Though I'm not going to try to name the differences because i don't think there's any way I'll get that described adequately such that we will agree.
I guess that was a long way of saying 'choose from multiple options' isn't really an adequate description of this play.
Honestly, I don't think it has anything much to do with 'choosing from options'. I think if we are going to really elucidate this whole aspect of agency we have to look at the ACTUAL differences in the two types of play, primarily trad (mainly simulationist, though variable, but always with setting/situation as a focus); vs Narrativist, with characters as the focus. When I say focus, I mean that which is the guiding element or originating impetus in terms of what fiction is deployed into the game next.
In Trad play there is an environment, which is prepared. It may not cover everything, and it may contain varying amounts of meta-plot, etc. but it lays out environments and situations; maps, rooms, NPCs, plans plots and motives, and usually envisages a general course through which play might move. Again, the could be more or less fleshed out, linear, non-linear, or beyond non-linear could be designed as a 'sand box'. Players may well be allowed to assert certain amounts of control over the fiction beyond their mere character's persons, or not. Techniques may be employed, or simply a social contract, under which the action goes in a specific direction, or the GM may be expected to improvise or author additional situations and environments whenever the PCs 'run off the map'. What is most characteristic however is that the focus is on these situations/setting/environment. The play loop is basically GM revelation of fiction, players declarations of actions intended to mitigate any threats and explore the current location, followed by GM revelation of whatever new location is moved into.
Character-focused narrativist play consists of the GM reflecting on the character's goals, situation, relationships, etc., discerning what will put pressure on them, and describing that (all with regard to past fiction, possibly drawing from prep where that is useful). The players then enact the needs and drives of their characters by addressing whatever the pressure is (IE usually an obstacle). Once this is overcome, the cycle repeats. Setting/situation/environment are tools, which the GM will employ and which the players can reference in describing their character's actions. This fiction is important, as it provides structure and constraints on the action declarations and what the GM can declare, but there's no goal of enacting any certain fiction. There's no specific situations that are anticipated to come up or need to be enacted to drive the story forward, except as part of the above loop (and thus how they impact the PCs and their needs/drives).
Both types of play can take into account things like genre, tone, desired themes and agendas.
One test you can use which will often distinguish the two types of play is to ask whether or not you could produce essentially the same outcomes if a different group of characters was used. So, as a fairly simplistic example: The Tomb of Horror is clearly a trad (even classical, the most trad of trad) scenario. Regardless of which PCs enter within, they will have largely similar experiences. The goal will always be to survive and penetrate the Lich's sepulcher and gather Acererak's treasures. Different parties may have slightly different mixes of abilities, rogues vs wizards vs fighters vs clerics, etc. but they will encounter exactly the same challenges and probably overcome them, or fail, in very similar ways, only varying in detail at the level of what specific action or resource is employed in each case at a given juncture.
Now, let us imagine a Dungeon World equivalent: A group of players has, through PC build choices and actions taken over many levels of play, developed and followed up on the tales of a lost Tomb. Finding this tomb and penetrating it embodies the final Dangers of the Campaign Front of the DW campaign! The redoubtable Paladin wishes to purge the evil lich from the world. The wizard wishes to obtain fantastic arcane knowledge with which to rescue his mother from a magical prison. The halfling thief doesn't like to admit it, but he's come to admire that stupid foolish paladin! The cleric has been commanded by his god to defeat the lich and recover the Staff of Ruin to atone for his past sins. There may be a partial map (with holes in it) which the GM has prepared to help him describe this place, but the story is about these particular characters. That map was created BECAUSE they have the specific needs and drives which I have described, it won't exist for any other unique group of high level DW PCs. The game play is not ABOUT the Tomb of Horrors, its about Robert de Morgan, Zorbilar the Magnificent, Harry Quickfingers, and Father Charlie.
And the process of play for each of these games, while similar in the same way that many board games are similar, is distinct and in each case serves its particular ends.
