Well, I think we're meant to take the writers wording literally, meaning Zefram Cochrane is from Alpha Centauri. They hadn't set anything in stone yet and it was purposely left vague so as to be open for interpretation; and it was just a throw-away line.
Cochrane was repeatedly and explicitly called human. The Companion created near-exact Earth conditions for him on the planetoid. He said "It's not Earth, but it's home." And he was uncomfortable with the idea of romantic attachment with nonhumans. Except for the single reference to Alpha Centauri, everything else in the episode creates the impression that he's an Earthman. Yes, he was known for being "of Alpha Centauri," but probably because he settled there later in life.
Now, maybe Gene Coon might've assumed that humans settled Alpha Centauri at sublight before Cochrane invented warp drive. After all, "Space Seed" assumed that the technology for sublight interstellar travel existed by the 1990s, and Cochrane was born 235 years before "Metamorphosis," so it kind of works out (as long as you go with a later time frame for TOS than "Space Seed" implied). But that's clearly not true now, given what
First Contact established.
And any fan theories about some human colony seeded on Alpha Centauri by the Preservers or something are just that, fan theories. There's no way Gene Coon had anything that convoluted in mind, especially since the Preservers wouldn't be invented for another year.