That probably originates from records, where one offs and limited releases are literally cut with a special lathe. I imagine most songs referred to as a cut are probably remixes or VIPs. There is a big culture of one-off releases for DJs. Look up the history of Dub Plates if you want to learn more.

nah it was cuz fantano was constantly using the word cut to talk about songs on an album ahahaha

Here's a solid video about it youtu.be/xKMt-TQJjc8?... talks more about the modern day continuation of it, but this sort of thing used to be the norm for dance music producers getting new music out to DJs back in the day before songs got full releases

Cutting It: A Dubplate Special | Resident Advisor

YouTube video by Resident Advisor