I can think of a way to end this debate really easily.
Okay lets think of a sci-fi series close to Star Trek, for this I'll go with Space Battleship Yamato since its kind of star treky even has the color coded divisions thing for its starship crew.
Alright does the crew from Space Battleship Yamato do basically the same job as the typical Starfleet crew? Answer: Pretty much.
Are they members of the military? Answer: Yes.
So case closed Starfleet = military.