I've been gaming for just about 25 years now, so I think my definition of the genres has become pretty strict over the years.
My definition of an adventure game is:
You play as a protagonist that exists in the game world.
You are either presented with a story, or you uncover it as you go along.
There can be action sequences, though I'd prefer if they were left out (I play adventure games to relax, not to push a lot of buttons)
You solve puzzles along the way. It doesn't really matter if they're inventory based or part of the world. However, and here's my most important distinction between puzzle and adventure, these puzzles have to be a part of the game world, actually be integrated into the world you're exploring, and not just there for the sake of having a puzzle.
I would definitely consider the Myst series adventure games. They were a bit atypical, since they didn't really have any of the traditional dialog we were used to. You didn't participate in conversations, you just listened in on them. Where the Myst games differed from other games at the time, were the style of the puzzles you solved. They were based very much on logic and math, and not the more traditional 'item A+B=result'. However, they were always integrated into the game world, and not just there for the sake of having a puzzle. Oh, and by the way, the Myst series has a very deep storyline, it's just not presented the way we're used to. You're not part of the story, but it's your job to uncover it.
A game like Big Brain Wolf is, for me at least, very much a puzzle game. While some of the puzzles you solve are loosely integrated with the task at hand, most of the time they're just there for the sake of having a puzzle. Definitely not a bad game, but doesn't fit my definition of an adventure game.
Anyway, that just about covers it I think
