Neonchameleon
Legend
100%It seems to me that there may be a profitable distinction to be made between:
1. We are creating story with planned direction - 'OK in the next scene Bob can meet his nemesis and then win after a close fight, but at the cost of his friendship to Jade and he gets a scar that reminds him of his past traumas'. We know what the story is going to be, at least in the very near term.
2. We are creating story without planned direction - the game is full of story-fuel or story-generating elements such as personal enmities. fragile relationships, opportunities to escalate at a cost, pyrhhic victories, psychological traumas, goals, flaws, etc, and we bash them all together to see what happens. We know a story is going to happen but we don't know what it will be.
3. We are creating story as a byproduct of other things - the characters fight monsters and explore dungeons and our focus is really on battle tactics or experiencing a fantasy world, so story isn't really on our minds. But, sure that will constitute a story of some kind I guess, and you can tell it afterwards with more intentionality if you want.
I think a lot of games that might be in box 2 get characterised as being in box 1 by people who don't really play them.
I would also say that you are also condensing two axes - collaborative scene building and pre-plotted arcs.
For example someone running Pathfinder: Rise of the Runelords (either edition) has vastly more of the story pre-plotted than any game of Fate I have ever seen and I would argue that any game of Apocalypse World or Blades in the Dark can.

