"Phil. Natural Spin Turn .Step four is a Natural Pivot. Half a Turn on the ball of the left foot for the man"
Not really. Yes, it appears to say that, but then look at the lady's footwork in the natural pivot turn and you will see that in fact the spin turn is calling for something less than the fully executed half turn pivot in the natural pivot turn.
The reality is that most pivoting actions are fairly lazy about the rotation, more concerned with sending both centers on as straight a common line of progression as possible, with little to no side-to-side deviation to get past each other. The forwards action is very much forward, forward, the backwards action just sort of happens, some pivot on the ball of the foot but also a lot of the body turning more than the foot.
Only if you are going to do something like a pivot into a lady's heel turn would the kind of completely turned action most nievely think appropriate for a generic pivot actually be appropriate.