Exactly, the music is different enough that the entire production counts as an original work, whereas the Glee version has neither, and thus would be considered a cover.
(edit: removed the happy birthday analogy, it doesn't really clarify anything)
In John Coulton's case, I'm not sure what he was required to do in order to license the lyrics for his original cover; i.e. whether he paid the calculated rate or negotiated something else. But it looks like Glee owes to two parties at best (if they actually lifted his audio like Timbaland this could get ugly)
http://www.slate.com/blogs/browbeat/2013/01/18/jonathan_coul...