You are correct. Traditional straight up Adventure games are like the above. More modern examples would be Hotel Dusk: Room 315, Indigo Prophecy, and Zack and Wiki.
Zelda, however, is Action/Adventure. The action elements are obvious enough. The Adventure elements spawn from using specifically purposed items to solve puzzles, exploration, and inventory management. I'd lump games like Castlevania II: Simon's Quest, StarTropics, Fester's Quest, and Okami into this camp.
Diablo is, by definition, an Action/RPG. As are games like Marvel Ultimate Alliance, Secret of Mana, and Final Fantasy Crystal Chronicles.
So the next question is how/why is Zelda Action/Adventure and Secret of Mana Action/RPG? The differences are subtle but look at how combat occurs, it's heavily stat based, features a much more robust item management system, and gives you a variety of equally viable options to defeating foes. It's also decidedly more storyline driven.
While Diablo has a pretty shallow story (Diablo II corrected this somewhat); it's system of inventory management, character creation/unique abilities, and shop system drop it squarely in the Action/RPG camp.
Puzzle Quest is uniquish, but it's still just a Puzzle game. It does feature a lot of RPG elements (stat/level increases, itemization, grand story line) but it's ultimately just a method to get you from puzzle to puzzle. But if you were to call it a Puzzle/RPG, I wouldn't object.
Another side project from this thread would be for someone to come up with a "master" genre list. The core set of game genres that ALL games would primarily fit into.