and that seems like a very, very different thing than theater
Well, and what I meant was that if I were to map my fave rave game/play activities on that line with, say, chess at +1 and Hamlet at -1, I think I would wind up clustering between 0.5 and 1--and then having that big red dot on the -1. Which usually means that the model is wrong for my perception of the world.
On the other side of the topic, I think that for the audience/fans, a basketball game very much creates a fictional narrative (if you believe, as I do, that momentum is essentially fictional). I don't know how much that is true for the players; probably quite a bit. But the players are only incidentally creating that narrative; their goal is not to amuse the fans but to win. The owner's goal is fan-amusement, that's the reason they are there, but they are, we hope and believe, ignoring that goal and concentrating on the goal of winning. This is very different, as you say, from the actors in Hamlet, who are supposed to be focused on fan-amusement (for varying values of amusement); the actor playing Claudius is not trying to win the narrative. So, sure, I see that as an axis. But that's in an audience-focused frame, not a player-focused frame. And in a taxonomy of play, I would use a player-focused frame.
And then the blocks…I suspect that my son makes games of his blocks, rather than toys. I vastly prefer games to toys, myself—if I am building with colorful blocks I will consciously impose rules on myself (make the tallest possible building from a given set, or don't allow two blocks of the same color to touch, or make the ricketiest possible structure, or don't allow setting aside of blocks for later, uswusf) which may not be obvious to anyone watching, but which I could fully articulate if anyone were willing to listen. I suspect that M. could not fully articulate his rules of the moment—he is building this way and not that way. But yes, game rather than toy, as he's using it. I think.
Taking this back somewhat to LARPing, wondering how this toy/game/play idea works out. Take a ren faire: there are people who are performing as (employed) actors in plays with a fixed narrative; people who are playing fixed characters without fixed narratives (perhaps demonstrators, vendors or regular visitors), people who are playing with the idea of renaissance-y characters without any fixed purpose much like your putative basketball-toy-er, people who aren't participating at all… do you feel that your axis helps to model that universe?