My guess was that Quirrell was already possessed by Voldemort, and that both events happened to Quirrelmort roughly as he described, the first with him acting under the identity of Quirrell and the second with him openly identifying as the Dark Lord.
Update: This post has also been superseded - new comments belong in the latest thread.
The second thread has now also exceeded 500 comments, so after 42 chapters of MoR it's time for a new thread.
From the first thread: