...except for the fact that you're arguing a moot point. You're arguing that *real people* call a *place* "Wake Island" because "Wake" is a generic term. So far so good, makes sense - you've got me convinced... But that doesn't have anything to do with the subject at hand.
That reasoning *must* not apply to the carrier, because
FA tells you that a *WC pilot* in the 2660's calls the *carrier* "Wake" - generic term or not, that's what he does and what he seems comfortable with. He didn't say "the TCS Wake", "the Wake Island", or "the TCS Wake Island", which going by your logic, is what you'd expect him to do.
If anything, your logic is reversed when it comes to the carrier. Even though there's no evidence pointing to this direction, let's assume for a minute that the carrier actually was named "TCS Wake Island" by Confed (also assuming that the WW2 "Wake" is the place Confed meant to make a reference to). What then? Well, that doesn't change anything.
FA still tells us that "people" still call the carrier simply "Wake". Not only do people feel no need to tack "Island" to the end of the ship's name, but rather, they might have removed that part from the name in everyday speech if it was present in the carrier's official designation (perhaps because they don't know of such a place or historical event or just because they might have shortened the carrier's name since they refer to the carrier in their daily lives more commonly than to an island that's half a universe away).
15-word version: You're arguing that people wouldn't call a carrier "Wake" without saying "Island", when that's
the only thing FA tells us about the carrier.