"Normal Pivots in the Standard style require that the front and the back foot be kept in CBMP."
Yes, but what is CBMP, literally? It is a foot position - it says nothing about the orientation of the body relative to the feet. Customarily, in the classic outside partner and promenade usages of CBMP, there's definitaly an associated opposite-side-lead position of the body. But not in pivot... in fact there is the opposite of that, a same-side lead.
"I seem to remember that you said that in a Spin Turn you didn't Pivot with CBMP, you stepped around. Was that a correct quote. Would that create the sideway step you mentioned, that you seem to think exists."
The action of a spin turn is not very much like that of the completed pivot you would dance if you were going to follow it with another pivot, or a true fowards man/backwards lady step (cf. running right turn). Instead, it's something between a pivot and an ordinary pointing alignment such as seen in the back half of a plain natural.
If you were - as we often do - to dance a series of pivots ending in a spin turn, there'd be a change of technique in the last - instead of holding the kind of same-side lead that sustains the rotation, the shoulders would be allowed to swing through into an opposite side lead.