To me, it will always be Constitution-class. It was referred to as such in "The Naked Now" (TNG) and "Relics" (TNG) as well as Star Trek 6... That's 3 referances against the 1 reference to it being an Enterprise-class (which could be a nickname for a certain class of people). It's also firmly already referred to as Constitution-class (and since it was said it was a redesign and refit job in TMP, a Constitution-class [Refit]) I never thought Enterprise-class suited the Enterprise... Too egotistical for the ship and her crew robbing the title of first ship in the class from U.S.S. Constitution due to it's famous crew from 2264-2269.
Going at this at a more opinionative state, I think deeming the U.S.S. Constitution and her sisterships, even the U.S.S. Enterprise, not Constitution-class Starships throughout their entire services is like saying the NX-01 from Enterprise is of the same timeline as TOS or TNG would be, when it has so many inconsistancies that it is basically another timeline and reality all-together (I.E. "Parallels" [TNG]. 285,000 hails from 285,000 Enterprise-Ds came to a single Enterprise-D... Meaning there was 285,000 realities. Suppose one of them were a more-advanced reality when Phasers and Transporters existed in the 22nd century, and so was Starships like the NX-01 fitted as well as the NCC-1701 Constitution-class Enterprise for exploration missions and in Nemesis, we see a glimpse into the timeline that included the NX-01 Enterprise in it's timeline [Mabye Data and the rest of the Ent-D/Ent-E crew existed after Nemesis in the timeline we know before the NX-01 was even thought of!], which all sums up to my Parallels Theory) ... And this comes to the fact that we don't always go by canon material. Pestalence and I don't always go on canon material, but on different things that is. This is one of those times where he and I would disagree.
I hope I didn't confuse anyone... I think I'm starting to confuse myself, reading this over and over again. lol