I always took it that Zephran Cochrane was a human from the Earth colony at Alpha Centauri...Humans could reach Alpha Centauri without warp drive as it's the closet star to Earth. So it's logical Rodenberry choose Alpha Centauri as Cochrane's home to show that humans had started to migrate out into the stars. It could be said that Zephran Cochrane invited the warp drive on Earth and later moved to the Alpha Centauri colony. That's the idea I'm going with.
Part of what you said was indeed relayed in the book (and I'm not saying it's the ST Bible or anything, just that it was written in concordance with the first film, Roddenberry was alive to contest it for over a decade after it was written and never did, but the retcon only occurred after he was dead and couldn't say if it didn't follow his continuity whether he wanted to or not - I always thought that was a bit sneaky of the writers and why I continue to talk about it!).
In the Chronology (which I still have, but it's in storage so I can't reference it directly) the history is written as ship logs, science journals and news reports (as if these references were compiled into one resource book).
I believe the Earth ship that reached Alpha Centauri was called the Icarus - it was a sub-light vehicle (we'd gotten to the point where we could travel ALMOST at light speed, but hadn't made the warp breakthrough yet.)
It took the ship several years to reach the nearest star (since it couldn't go the speed of light and A-Centauri is 4.5 light years away). Don't remember the exact number of years cited in the book or if they used suspended animation for the trip or just had a long journey aboard a big ship. They obviously had suspension technology since Khan's people used it as early as the late 1990's.
Once there, it was the crew of the Icarus that made first contact with aliens and met Centauri native Cochrane. The logs explain how language was first worked out so the two species could communicate, then the ship's scientists began working with Cochrane on his warp theories & prototypes. The Earth ship was on Cochrane's planet several more years as they collaborated on building the first warp engines - when finally done the trip back to Earth was much quicker than the trip there due to the new warp engines.
The authors based these events on info from the TOS episode Metamorphosis and apparently with the approval of Roddenberry. But the story was radically changed with First Contact.
Except for Cochrane in the TOS episode, I don't know of any other depiction of Alpha Centaurians as an alien species on the TV shows or in the films (of course, I'm not an ST expert, either!)