I jaunted over to TrekCore and compared the McCoy-in-the-Defiant's-Sickbay scenes between SD and HD. The one clear HD scene where the camera pans across Sickbay's convalescent ward has a fairly clear arrowhead on the one man laying in bed toward the end of that scene. TrekCore's SD slate doesn't include that particular frame, so it's academic for me.
The next scene where they show the operating room/examination room also shows a man slumped face-up on an examination/operating table, but its distant enough to be academic for me. It's fairly clear in HD, though not nearly as good as the previous pan-shot.
For me, it can go either way. If the Defiant's crew had Enterprise-style arrowhead badges, that means both ships are attached to the First Fleet. If we're assuming they could be something else, whatever it could be, then either Defiant is in another flotilla (orbitilla? warpilla?) or we can still pretend that each TOS ship had its own distinct badge, whichever fires your phasers. I prefer to think that it doesn't matter which badge the Defiant crew wore; there were enough examples that badges indicated (to me, anyway) some unit assignment (such as Kirk's arrowhead meaning the First Fleet).
It's probably just me, but I would've preferred if the TMP and TNG ships had retained the TOS boomerang-style Starfleet pennant on their hulls, and a diversity of badges on their uniforms. It may have looked militaristic to some, but today's NASA astronauts seem to be quite a bit less militaristic and they seem to have quite an interesting variety of patches just the same.
Fires my phasers!