I actually think the first U there does stand for "United Earth" in Archer's time, whereas in Kirk's ship it stands for "United Federation of Planets". It's just that both the S'es also have a raison d'etre: the first is "Star Fleet" and the second is "Star Ship". UESFSS would be a mouthful, let alone UFPSFSS, so the shorter forms get used.
(In Season Four, didn't we get some display graphics showing "USS Enterprise
" despite there never being extra letters in the actual pennants?)
As for the "NX-Alpha"/"Constitution
-Alpha" thing, it's a bit unsatisfactory in this respect that the great project actually usually gets called the Warp Five Engine Project rather than the NX Project. I mean, it makes excellent sense that the test rigs would exist solely to test the engine, while the greater project would involve lots of other testing and the warp rigs we see would actually just produce the generic engine that could then be applied on just about any ship class. But for some reason, the test rig still gets called NX-Alpha, rather than something more logical like Archer-Alpha or WF-Alpha...
Since the very concept of an exploration starship was so unique at the time, though, I'm willing to believe the engineers would see even the components of the project in terms of the end product starship. It might be too much to start believing in a general revolution of starflight; the Warp Five Engine might live or die with the exploration starship project, regardless of whether the engine was a success or not. Rather than jinx the general revolution, the propeller-heads thus only speak of the NX project specifically.