It does appear that Paul was only a Libertarian for a few months.
But that does not contradict anything in the narrative above.
Paul briefly contemplated an independent run in 1992.
Paul never comes out and says that he was prohibited from running as a Republican, but he did say it was for "personal reasons".
The behind the scenes jockeying is never 100% clear, but there is other evidence out there that the Party would not allow Paul back into the fold as a candidate.