Does Button Rhythm change speed on fades or moving shots?

Yes, and you do not set these yourself. The script applies context multipliers to the sweep depending on the shot: turbo fades sweep about 18% slower; insta-stop and quick-stop shots about 10% faster; shots on the move about 8% faster.

Shots taken within 400 ms of leaving the move stick — a step-back or hop gather, for example — also sweep slightly faster. Your Button Rhythm value stays the base; the multipliers adjust from it.

Source: yew2K mod menu (shipping GPC).