:::: @RQG: A point you're still missing is that there is more than one actual mathematical definition.  Different characterizations of a concept can be equivalent to each other, and which one is taken to be "the" definition may be a matter of convenience varying with the context.  As I pointed out above, one of the characterizations (as "(A,V)") superficially makes it appear that an affine space has more structure than a vector space (since the vector space is only "V" rather than "(A,V)").  But I explained why that is a superficial appearance, and that particular way of defining the concept of affine space is not the only way and is not sacred.