Earlier quoted context omitted.
PC originally meant "IBM PC desktop clone". Apple was different because it's hardware wasn't compatible with PC software (for mainstream users).
"PC" was "personal computer" years before the IBM PC. Just look at old computer magazines. The whole reason the phrase "IBM PC" existed was to differentiate it from the other PCs that already existed. "IBM" was the adjective. "PC" was the noun. Because of its success in offices, "IBM PC" became just "PC" the same way other words like "omnibus" became just "bus" because it's simpler to say.
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.)