"Not my theory: I'm talking about the standard technique of the teaching society of which I am a member."
No, you are talking about a collection of historic mistakes and misunderstandings of your society's text.
"As for Step 4 of a Pivot Turn: it is exactly the same (as man). Only the man has a pivot in either turn."
Because it is substantially different for the lady in each case, the man's accommodation of her will necessarily make the detailed execution of his corresponding action different as well.
Your obstinate insistence that step four of both figures will be exactly the same as man is a perfect example of what happens when people let slavish adherence to an over-simplistic reading of the text take priority over essential dance skills such as listening to your body and accommodating your partner. Where you instead to employ those in your dancing, you would find that the actual execution as man of two things that look the same on the page turns out to be practically different.
Fortunately there are officials within the teaching societies who understand both the purpose and the limitation of the text, and are able to carry on enlightened discussions of how to do dancing that honors not only the fraction of the traditions represented by the ink on the page, but also the more complex demands of partnering and musicality at a level of sophistication which cannot be captured in a chart.