I have one nitpick: You say, “We have to take the ratio between two utility differences”, but really, because only positive affine transformations are OK, what we really have to take is the ratio between a utility difference and the absolute value of a utility difference.
Why? Positive affine transformations are OK, and they don’t affect the sign of utility differences.
Why? Positive affine transformations are OK, and they don’t affect the sign of utility differences.
Yes; the point of making this change is to exclude negative affine transformations.