As I recall, the most widely used term early on was "microcomputer", to distinguish the small home computers from the larger "minicomputers" (e.g. the DEC PDP-11 of blessed memory). The term "personal computer" was also in use, but the use of "PC" was not common until after the introduction of the IBM model 5150 (whose actual product name was the "IBM Personal Computer").

Since then, you might refer to any of a variety of machines as "personal computers", but "PCs" only meant "IBM PCs" (or later "IBM PC-compatible machines"). In other words, I would argue that the term "PC" derives specifically from the "IBM Personal Computer", and not generically from "personal computer".

Source: I was there. :-) I haven't done the research, but I bet if you did search through Byte and similar magazines of the time, you'd find plenty of supporting evidence. (I do have a memo cube from the early '80s with the slogan "Apple II -- The Personal Computer", but I suspect that was Apple Marketing trying to fight an ultimately losing battle.)