"A side-leading step is the opposite of a step with CBM, not of one taken in CBMP"
Wrong.
In both CBMP and side leading situations, the orientation of the body relative to the feet is pre-established. If you take a series of diagonal steps with unchanging foot alignment and body orientation, they will be in side lead, CBMP, side lead, CBMP, etc - CBMP and side lead are essentially the same condition, but the appropriate name depends if the if the moving foot is on the side where it will extend the body alignment or if it is on the side where it will step across the body alignment.
Contrastingly, in CBM actions, the orientation is established by body rotation during the step.
A feather is a great example of the typical sequence of events.
1) Use CBM to establish an orientation of the body relative to the feet.
2) This orientation means the body orientation is on the same side as the moving foot, so we say we have a side leading.
3) We are now moving the opposite foot across that same body orientation, so we now call it CBMP.
"We are very careful indeed with shoulder movements, aren't we? It sometimes helps beginners to feel a backwards CBM as coming from the hips, rather than from the shoulder (Moore), but that terminology dates from a time when Moore was still writing about "shoulder-leading", and its been a long time since anyone else did. Otherwise, by and large, we don't move 'em."
I would say shoulder rotation is substantial and obvious in natural CBM, but needs to be prevented during the early stages of reverse CBM. Some just simplify to saying that reverse CBM happens late, but that's not really true - what is true is that its effect in the upper body is later and smaller.