If a person is co-opted into a group, is that person necessarily a member of that group?
The only possible answer, in my opinion, is yes.
He was co-opted into the Labour Government in 1964.
The only logical inference, in my opinion, is that he became a member of the Labour Government. (Would it be logical to think he didn't?)
Once again a lesson on how important context is.
The word co-opt is not an everyday word. Unless you have a good reason to use it you shouldn't concern yourself with it any further.
(Please note that I arrived at my opinion only after getting a two-hour nap and reading everything that had previously been posted.
(If you make your post too long by the time I get to the end I may have forgotten what was at the beginning.)
Your response indicates the Cobuild definition is inaccurate.