Then how do you separate games like Myst from puzzle games like Pandora's Box? Myst did have a story -- you had to find out which brother was telling the truth and figure out what really happened at that deserted island. And Myst had exploration of the gameworld, which Pandora's Box did not. Just because Myst had no conversations and was non-linear doesn't mean it had no story.

I don't mind classifying Myst as a puzzle game, but the difference between it and games like Syberia isn't whether there is a story or not. It's more about the types of puzzles that predominate in the game, how much interaction there is with other characters, and possibly about the linearity of the game. And you still have the problem of differenting between games like Myst and games like Pandora's Box, which aren't much alike at all.

Why might one want to classify games? Depends what one wants to do.

You might be interested in knowing how many games were produced each year, perhaps to determine how well the Adventure game genre is doing. It'll be nice to separate out indie / freeware games from more commercial titles. While you could do that by tracking down reviews that's a bit long-winded.

Perhaps classifying games is a bit like stamp collecting. I don't collect stamps myself but good for those who find it a rewarding hobby.