I think it's useful to divide the concepts a bit:
1. Players control characters. DM controls the world. Both can react and act on one another, but can't control the other.
2. Players control characters. DM controls the world. Players can spend resources to modify the world directly without character action.
3. The characters and world are controlled by all players. Players might have specific characters or parts of the world they control.
Now the question is, are any of these categories strictly **not** Storytelling capable? Are any of these categories better or worse at it? Why?