> The term I was looking for was affine structure, as I commented to someone else. But from your link, which I can't understand entirely, I get the sense that a torsor is an even bigger generalization.

An affine space is a torsor under a vector space, and you can have instead a torsor under any group. This loses a bit of structure, in the sense that you can take convex combinations in an affine space but not in an arbitrary torsor; but otherwise it is a proper generalisation. But the convex combination $(a + b)/2$ used to obtain a midpoint is exactly what we want here!