See, that is what I, personally, would disagree with. Like I said above, the genre name comes from specific games that had "adventure" in the title, but it is neither a requirement for adventure games that you have to go on an adventure (they could also take place in an office), nor are all games in which you go on an adventure related to adventure games (e.g. Rick Dangerous).

I grant you that it's hard to imagine an Action-Adventure in an office though. ;)

The assertion that the genre name was caused by specific games putting the word adventure in the title is debatable & unprovable. One could just as easily claim that the name came from the fact that you go on adventures in Adventure games, and that is far more likely to be true.

If you don't go an adventure, then no, it's not an adventure game.

Yes, games where you go on an adventure are not necessarily adventure games...but that's why describing what the player does is only one element of what genre a game goes in, but not the whole formula.

Well, I can only refer you to this post. We'll have to agree to disagree. In any case, the theme "adventure" in adventure games is much too broad and vague IMO to be of any help. It would be easier to list games that don't have the theme "adventure" than the other way around, so as a defining criterion (even as a descriptive one) it's rather useless.

Personally, I think the definition of Adventure game isn't the inclusion of story, but more that it's a game where interactions are more-or-less scripted. In particular, you don't have puzzles that follow a specific rule set (there are puzzles in adventure games, but they're more of the type "use this item to progress", not "push these blocks so that they form squares, which will then disappear"), and there's no combat system in pure adventure games either. As with puzzles, you can have combat in an adventure game (I actually made a topic about that some time ago), but there's no combat system. (The topic even had mention of a game where combat consisted of verbal debates, if I remember correctly; the game involved rabbis IIRC.)

The genres that are defined by a focus on story are visual novels (where story is the focus, with very little gameplay) and kinetic novels (where there's no gameplay at all).

Counter-examples here (where the game takes place entirely in one mega-dungeon (as I call it), but which would not be metroidvanias:
Wizardry 5
Etrian Odyssey (and other Wizardry-likes that have just one dungeon)
Ys Origin

Of course, going the other way, I could mention
Castlevania: Order of Ecclesia
(which appears to be a metroidvania, but has separate areas accessible by world map)

What about kinetic novels? (Take a visual novel, remove all the gameplay, and the result is a kinetic novels.) Are those still adventures? (If not, then gameplay does matter for the definition, as the only difference between visual and kinetic novels is the presence or absence of gameplay.)