chrisweekly parent
Huh. I'm from the Boston area, 50yo, avid reader, former English teacher, fairly well-travelled...
I've definitely been using phrasing like it ("save off a copy", etc) for at least the past 20-25 years (upstate NY, moderately online, avid reader, parent is an English professor, fairly well-traveled).
Conceptually, I think that the "off" serves the purpose of aligning it with something like "split off"—you're essentially forking the history by creating a separate saved copy.