Well this is just absurd denial of obvious proof.
I'd say, rather, that what needs to be considered is not who "installed" the floor emblem in the fictional universe, but who conceived of it in reality and what their intention was. Clearly it was an attempt by the show's art department and/or producers to finally offer a reconciliation between UESPA and Starfleet and answer a question fans have been wondering about for decades. I think a lot of people fail to keep in mind that most of the people who were making
Enterprise by that point
were longtime Trek fans just as much as any of us. They'd participated in the same discussions and debates as the rest of us, wondered the same things as the rest of us, and imagined possible solutions for the inconsistencies the same as the rest of us -- and once they were actually writing and producing the show and designing its sets and graphics, they were in a position that most of us would envy, a position that let them actually incorporate their answers and plot-hole fixes into canon.
So I don't understand why anyone would object to the fact that fans just like us took the opportunity to sneak in an explanation for the UESPA/Starfleet question by putting both names on the same emblem. If anything, I'm surprised they didn't do it sooner. (Honestly, if I'd been in charge of
Enterprise, I would've had the ship answer to UESPA instead of Starfleet. But maybe that's one of those cases where the network insisted on including more familiar Trek elements.)
UESPA had its own military, the Starfleet. It makes perfect sense for the Starfleet to be originated from NASA-like scientific exploration organisation, sure it is also the main military arm of the Federation, but it is much more. They are not only soldiers, they're also diplomats, scientists and explorers.
Keep in mind, though, that the United Earth Starfleet presented in ENT was not actually the UE's military arm at all. NX-01 set out with a goal of pure exploration, and when the crew found themselves encountering combat situations more often than they expected, they actually had to refit the ship to install heavier weapons. When the Xindi attack happened, NX-01 was sent out to respond to it because it was the only ship fast enough, but it had to take on a new contingent of specifically military personnel, the MACOs (basically space Marines), to do the actual fighting. As late as season 4, Captain Hernandez said she hesitated to include MACOs in her crew because she wasn't comfortable having military personnel aboard her ship. So even after the Xindi crisis, UE Starfleet was still considered a non-military body, despite using military ranks.
The way I see it, the way I've gone about it in the books, is that the Federation Starfleet is different from the UESPA Starfleet because it's a hybrid of different worlds' space services. It's a joint exploration-defense force that includes UESPA as its exploratory arm, the Andorian Guard as its defense arm, the Vulcan Space Service as its research and scientific arm, etc. Over time, these functions become more integrated and we get more multipurpose ships and crews that combine all the functions.