To me, "obvious" and "outward" are two different things (yeah I know, self-contradiction there), whereas "flaming" is just another word for either "outward" or "obvious", but "flamboyant" only applies to gay males, and means both outward and obvious, while also being effeminate and garish.
"Obvious" meaning that most anyone can read you as gay, and "outward" meaning you really want people to know and just make it a big thing about you.
I'll call myself flaming because I'm extremely outward . . . though not sure if I'm obvious since I've never really been considered a girl to begin with (outside of the trans community [there are a myriad of trans people that are LGB as well] and corners of the internet where gay people flourish, so it's no surprise if I am anyway).