you're right on that.
The FPA is a brach, as you said, that manages to create a root on it's own thanks to the main differences between FPA and FPS.
- FPA is escencialy a seamless world while FPS is a collections of stages using cutscenes to connect them.
- FPA revolves on one main, big objective while FPS uses a bunch of small objectives to clear the stage.
- FPA can have backtracking, you can explore a world that you cleared and you can see the effects that you made. In FPS you play the stage like you never cleared it.
- FPA is more open while FPS is more linear
- FPA is about explotarion, FPS is about action
There are many things to consider, because FPA's have bigger challenges than FPS. But when they are beaten, like in Metroid Prime, the result is a great game :)