I don't have a historical source it was just a term we used to use 10+ years ago when I started transition. We used to talk about "going full time" etc.
It's pretty intuitive a concept. A part time trans woman is only dressing and living as a woman in some parts of her life but not 24/7. She's cross dressing to gay bars on the weekend, or maybe whenever she goes to work or sees family she dresses masc and tries to look like a cis woman.
I'd say it was pretty rare to do like, the Dylan Mulvaney thing and just immediately start identifying as a woman in all social contexts before you'd done any changes to your wardrobe or hormones. I think it was more common in my observations for people to start their medical transition and wardrobe transition while part-timing and then once they reached a critical mass of clothing, hair length, boob growth etc. then they'd start presenting full time. This is what Abigail Thorne famously did. It results in the transition looking more abrupt to outsiders than it really was.
People don't really talk about it anymore but a lot of people still do it. Idk it's just a helpful term for describing a thing that people do. I was part time when I was still figuring out what gender presentation I wanted to go for so at school I was using they/them and they/she but at work, with family, on Grindr, I was living as a gay man. Then when I got on hormones I started full-timing as a woman. By the time you're full-timing people are usually getting suspicious and noticing that something is up that they can't explain.
It's just a part of transition and I think more people still do it today than we recognize.

