If we are to put this into real world context, then we could argue that UE Starfleet is taking all its cues from USN practices.
I mean, the writers probably don't do that, not really, not consistently. But Starfleet would be somewhat likely to. And in that case, the "NX class" thing would be pretty clear-cut:
1) Every ship design gets called a "class" and is named after the first ship commissioned, with few and minor exceptions (sometimes it's the first ship launched, or whatnot). Hence, Archer flies the Enterprise class of starships. A real world, USN counterpart example might be the "Ticonderoga class" of anti-aircraft cruisers.
2) Every ship gets a pennant code reflecting her mission. Several designs may perform the same mission. However, here Starfleet explicitly has never operated a deep space explorer before. Hence, Archer flies an NX class vessel, and prides himself on the fact that his class is the only NX class in Starfleet, even if there are other individual ships in this particular NX class. A real world, USN counterpart example would be as above - the Ticonderoga was the first "CG class" vessel in the USN for a long time, and also the first-ever "AEGIS class" vessel (a designation deriving from her pathfinding fire control system).
3) Every ship also falls on a broader category that may encompass several parallel mission classes. Say, there are many sorts of destroyers, some optimized for anti-aircraft work, some for anti-submarine, others for shore bombardment jobs. Multimission capability is common, but emphasis may be placed on some mission over the others. Hence, Archer flies a starship class vessel, or perhaps a cruiser class vessel. A real world, USN counterpart example might indeed be "cruiser class", as although it's not a broad category there nowadays, it used to be common yet encompassing ships with highly dissimilar capabilities.
The three ways to spell out the class of NX-01 are in no contradiction with each other, and all derive from real world practices. There are further ways, too, such as nicknames ("688 class" for a distinct type of nuclear attack submarine the first of which happened to have the registry SSN-688) or thematic names ("County class" for a class of Royal Navy ships named after counties, there being no "HMS County"). Heck, for all we know, this "starship class" thing that pops up now and then in connection with Kirk's ship may be of the latter sort - the NCC-1701 and her sisters are thematically named after past starships!
Of the above, only the second thing would fail to apply to other hero ships, from the UFP era. Both because the Starfleet of the UFP doesn't spell out the mission class in the registry, but uses the generic NCC there to merely denote "this is a Starfleet vessel" and thus separate the ship from civilian counterparts with letters like NSP or NTD - and because there's little or nothing distinct about the missions of later starships...
Timo Saloniemi