You are asking for a short game with a story. I'm still not sure if it's clear enough, I fear that you might have a lot of game submitted that might skip that requirement since the name of the jam is "the shorter the better" which goes a bit against making a game based on story.
What kind of game would qualify then?
I'm going to rewrite the basics a bit to make it clearer.
The idea is to make a game that can be played in 5 minutes or less. Within that limitation, it should explain something. The important thing is that the narrative drives the game, but if not, there should be some story. We're not going to get too serious about how the story is told. What we want to avoid is it becoming a game jam of "purely mechanical" games with nothing to say.
We also don't mind if there's little "gameplay" (like in a "choose your own adventure" or a "choices matter" game).What kind of game might be excluded? Well, honestly, I couldn't say. Because you can tell a story with a small puzzle.
When the film industry talks about "a short film," officially they mean a 3-minute movie (although this is often exceeded). It seems like you can't tell a story in 3 minutes... but you can.
If the game lasts, say... 8 minutes, well, that's fine too. In the end, "The Shorter The Better" is just that... "the better." We insist on the 5-minute limit because we want it to be a challenge.
So... what kinds of games qualify? That said, well, any kind.
I think I'll write an FAQ section.