...Clearly, all other parts of the United States defense establishment have abandoned their traditional names and most of the other pertaining traditions as well in the founding of this Starfleet thing. Probably in stages, so that Archer's Starfleet may have absorbed a couple of earlier United Earth defense branches, which in turn have absorbed the national defense branches, in an unholy mess that left the Royal Navy survive long enough to be the employer of Reed's dad.
Something of the British naval tradition survives, though, including the rank scheme. And it appears to survive through its brief stint as the US naval tradition, so we have lieutenants rather than leftenants there... But generally, there doesn't seem to be any sort of national pride or unit pride that would have made the transition from 20th century fractured Earth to the 23rd century United Federation of Planets, or even to the 22nd century United Earth.
Both Vulcan and Betazed have their own defenses, implying that Federation members in general have their own fleets and planetary defenses.
There's nothing to suggest that these forces would be separate from Starfleet in any way, or even predominantly (or at all!) crewed by Vulcans or Betazoids. As far as we can tell, they just happen to be those parts of the overall force that at that time are engaged in defending the respective worlds. In "Unification", the E-D would be part of the force of Vulcan defense vessels spoken of; in "Take Me In The Holosuite, Hard", the Vulcan vessel
USS T'Kumbra was part of Starfleet and would have been part of the Bajoran defense force had said force come under attack during the events.
I am truly at a loss as to what the distinction is between "Federation Starfleet" and "United Federation Starfleet."
The only one that matters?
Remember that the question was
solely about what the futuro-space-jarheads ought to be
called. One side advocated "United Federation Marine Corps" for the sake of tradition, the other ridiculed this because the use of the prefix "United Federation" is not part of the Star Trek culture in any context. It would be odd to the extreme for one sub-organization to use UF when others exclusively use either UFP or just F.
On the other hand, the use of "Corps" in the title has some canon precedent from the Starfleet Engineering Corps mentioned in ST2:TWok...
Timo Saloniemi