Here is what I'm currently theorizing....
1. Role playing games are an aesthetic activity
RPGs are do not produce separate financial or material profits for players. The main product is purely based on aesthetic benefits -- though this is less likely to be about artistic aesthetics than it is to be about an aesthetic of pleasure/enjoyment/fun.
What Edwards proposes as a key axis of description is, in my mind, merely a matter of aesthetics, and one of several elements mediating sense-making activity.
2.Role-playing games are a collaborative activity.
Even in the mythic old days of GM autocracy, you can't run or play a game alone. Solo adventures and choose-your-own-adventure type games are a very different qualitative experience than tabletop games, as are solo computer RPGs. Part of the appeal of MMORPGs is in fact this element, though I would argue that they are different in terms of aesthetic and the rules governance of the activity.
This means every player brings their systems of relations to the table, including culture, language, values, aesthetics, and communication styles. Collaboratifve processes are another mediator of the activity, both on the macro level of the whole gaming community and on the micro level of each gaming group.
3. Role playing games are rules-governed activity.
More important than the issue of game mechanics, the rules that govern RPG activity are the informal or formal social rules, including division of labor (GM vs players, for instance), expectations of activity (what is an RPG), and group etiquette.
This is a third mediator of RPG activity.
4. Role-playing games are made sense of narratively.
I am purposely NOT referring to RPGs as story-driven, though I am not excluding that possibility. Rather, regardless of intent, I think that most story for participants will still arise from the sense-making activity of players and GM and a player after the game, sharing stories, speculating, and using selective memory.
Thus, stories will arise no matter the intent of the GM, even in a dungeon crawl, as long as the players are still involved. Verisimilitude and suspension of disbelief will provide stronger scaffolding for this process, but it will tend to happen in the end.