Literally every single point-and-click I've played has extraordinarily arbitrary puzzles, so that rule might not apply.

I'm willing to disagree on this definition of an adventure game. Yes I am disagreeing with Wikipedia too. It is somewhat... confusing, looking at it from a technical standpoint. For one, any game can be narrative-focused. Game genres are not defined by their stories, but by their mechanics and how the player interacts with the world using these. In an adventure game you're going in to explore  and unravel a world and its marvels. If there is a story then it's a plus, because everything catches meaning and makes sense.

You are presented with puzzles, yes, however those puzzles are not exactly related to narrative. Some adventure games might be clever enough to implement it as a way to add to the story. But those puzzles are just part of the world as an obstacle to test your knowledge of it. To see how far you've progressed learning the mechanics.

Let's give an example; the very first Legend of Zelda. No one played it for the story. If you read the manual that comes with it, yes, it has a story. But the game itself wasn't meant to draw you in for a story. The only story that would exist on it is your own, the one you'll tell your friends about how you got wrecked on a late-game dungeon for going in too early. And that, was quite an adventure. You discovered things unknown to you before, you struggled on your way. Exploration was led by curiosity. (A deeper story would be added much later on the TLOZ sequels.)

Often the obstacles presented in an adventure game are your lack of knowledge and ability. I don't know what's around the corner, but am I willing to explore? Am I willing to risk not having the ability to overcome what's next? Do I know how the next puzzle works? No? Have I missed something? Is there another place I might need to explore? 

All in all, adventure games are ones in which you are placed in an unknown world to conquer and discover. It could be linear, it could be open. You're limited by the unknown and your abilities as a player, or in-game mechanics. This can also apply to other games, but adventure games focus better on their worldbuilding and interacting with it. They're not just there to tell you a story, they're also there to be explored. 

If you take away a story from any game and get nothing from under it, then it's not a game. Videogames are interactive experiences after all.