I'm complaining that there isn't one uniform distribution. For example uniform in x on [1, 2] is different to uniform in 1/x on the same interval, and which one

I'm complaining that there isn't one uniform distribution. For example uniform in x on [1, 2] is different to uniform in 1/x on the same interval, and which one you draw from may matter. You have to be told. In fact every (one-variable) distribution is uniform in something! "pick a random point inside a circle" comes with an implicit "uniformly according to the ordinary euclidean metric on the paper". I guess it's a judgement call that this is obvious enough, but "a random chord" is clearly not... otherwise we wouldn't be here!