Let me start this post by saying I'm going to be blunt about differences, but that I'm not making value judgements. In fact, I rather enjoy 5e, and I enjoy a good sandbox/hexcrawl in 5e (good doing a lot of work), but I'm expecting when I do this that I'm largely going to be consuming the GM's ideas. This is fine -- clearly I find it entertaining enough to spend a large chunk of my gaming time doing this as GM or player. That said, I also enjoy other approaches and games, ones that are very different from 5e. This has given me a perspective on the differences in play between playstyles, and I largely find that people that haven't engaged outside the D&D-alike spheres to have unwarranted views on just what's happening in GM-centered games (by which I mean the GM has the majority of authorities over the rules and fiction of the game).
Yeah, I'm sure that every DM running 5e as a player driven sandbox is simply deluding themselves. In point of fact we're all just running DM-dictated railroads. All the poor players can do is wait for us to tell them things. (That's sarcasm.)
That's it's player driven, yes, usually they are. What happens usually is that players get to chose which parts of the GM's prep they experience with more freedom than in a linear game, where the GM chooses which parts of their prep you get to experience. The difference between a prepped sandbox and an actual player driven game is that the latter lets the players set all parts of the agenda of play -- they get to choose what play is about and the game is then about that. This isn't true in a sandbox, where what you want only happens if it's a part of the GM's sandbox vision.
This gets made very clear if you branch out into some other games that actually do player-driven play and not the watered down expanded menu version that is the typical D&D sandbox*.
*caveated with typical, because there are some atypical ones out there.
Not having explicit control of the fiction does not equate to having to wait for the DM to tell you things. There's a reason that advice like Yes And and Yes But exist. If the players ask me whether there's a blacksmith in town, then I'm inclined to say yes (unless I have an established reason to say no) but maybe he's been spending his days drunk in the tavern because bandits killed his daughter. Or there's a blacksmith in town and he's actually someone that PC knew while growing up (and I can let the player fill in those details).
It kinda does, though. This isn't a bad thing, but a required thing. The GM is the sole source of the setting material and the fiction outside the characters. If I, as a player, want to know something about the situation or the setting, the GM has to tell me. I then get to take actions with this information (which is often incomplete as the GM is keeping secrets) to find out more information. For example, if my character enters a dungeon room, I have to wait for the GM to describe it's features. I can then take actions to discover more, like maybe what's in the chest the GM just described to me. This prompts the GM to tell me more, and the cycle continues. There's nothing wrong or bad with this-- it's a very entertaining and obviously popular way to play.
But, there are games where the player establishes things about the fiction, and then those things are tested by the mechanics to find out what happens. In this case, the opening may be the same -- the GM describes the scene, which could be a dungeon room like above, and may include a detail about a chest because it's interesting set dressing. The GM, in this case, has no plans for the chest -- she doesn't know what's in it any more than the player does. The player established that they're going to search the chest for something they want, say a treasure, that is important to the character. This gets tested by the mechanics (or the GM just agrees), and, on a successful test, that's what's in the chest! On a failure, the GM complicates the character's life, usually by denying the intent of the task and introducing a complication -- the treasure isn't in the chest, instead there's note from the PC's rival taunting them that they've already recovered it! Back to square one for the PC.
This is an incomplete example of play that focuses specifically on how D&D is largely getting the GM to tell you things and how that can look differently in a different system. Preferences can vary on which you may like, but the distinction remains.
There's a reason player driven play styles exist. Did your daughter have the players come up with a goal that their characters were trying to accomplish? That's fairly important for player driven games (more experienced players don't necessarily need to be coaxed to do this, as they will typically do so on their own, but for newbies it's pretty much a must).
If what you're getting at is that WotC ought to publish some kind of play style guide, I could see that. Although there is a veritable wealth of information on the internet, admittedly, separating the wheat from the chaff can be challenging, especially for those unfamiliar with the topic.
Again, there's really not much "player-driven" play in 5e. Can't be. All of the authority over the fiction is vested in the GM. They may indulge players, absolutely, but they have veto, and that's absolute control over the thing. So, a "player-driven" version of 5e is still entirely at the discretion of the GM -- the players have no way to enforce anything or bind the GM in any way. Other systems do do this. So, when you say "player driven" with 5e, that means that the GM is indulging the players, usually by offering a broad menu of things to choose from. This isn't actually player-driven, though, it's just a more expansive choice set.
