"Anonymous. Your 4th paragraph is the very thing that needs to be understood.When you say You push sideways from the first step.
You are speaking from the man's perspective only.
For the lady there is that little description which says NFR on step one.
The push will come later than the man's."
No, the trajectory of the bodies must match, so the lady will push sideways to partially redirect her movement at the same time that the man does. If she did not, then she'd be dancing a 3/8 natural turn rather than matching his desire to dance a quarter turn.
This is a separate issue from the delayed completion practiced by the person on the inside of the turn.
"In laymans terms Let him pass first. Do not try to arrive at exactly the same time."
In the 90 degree turn figure being discussed, he isn't going to pass her nearly as much as he would in the 135 degree natural turn, so this difference is going to be smaller than it would be in a natural turn.
Note however that I never said she would arrive at the same time - what I said was that she had to push sideways in order to make a change in her body trajectory that matches the change in his. She can still arrive later, but she must make a matching change.
Probably the simplest way to understand this is to turn the figure around and have the man dance the inside action while leading. He's got to lead the sideways change in her trajectory by pushing sideways from his feet, while also letting his partner get there first.