The numbered Prime games involve exploration, item collecting and huge boss battles which is why most would classify it as an adventure. You're shooting, sure, but it's not the main focus. Calling it a shooter is like saying Oblivion is an FPS because you can shoot bows and wands and whatnot.
Hunters is more combat oriented and it's main focus is definetly the online multiplayer (which it does fantastically btw). I have no clue why people like to complain about it's single player, it's obviously not meant to be the most enjoyable portion of the game.