The CBMP is part of the definition of a pivot. If it is not there then it is not a pivot - by definition.
How would you define a pivot in such a way that it is distinct from any other turn on the ball of one foot? I have seen one definition that includes "keeping the inner thighs together" which would also work.
However! Since Alex Moore, and Guy Howard who uses a similar definition, wrote those words dance styles have moved on and competitively there is more concentration on progression rather than rotation. Most choreography nowadays includes a series of "pivots", though these are not true pivots according to the definition.
The problem is that we use words from our normal vocabulary and assign them specific meanings with subtleties that are sometimes forgotten or missed. Ideally we should invent new words to avoid confusion.
eg "Squingel" = pivot with CBMP "Spandlett" = pivot without CBMP
We could then dance a series of spandletts followed by a squingel without argument. :)
Pondering , discusing. What a waist of time. There are many to watch on Pivoting. One is find your way to Edita Daniute and see how the champions do it. You might be suprised at the turning in of the toes with the left side leading on the backward step. Pointing out just in case it gets missed.