What you describe has more to do with character customization or lack thereof. While it is a style choice, the gameplay and the story influences heavily what you can present as an interface. There is just stuff you cannot do easily in first person. Not with players enjoying the game play. And vice versa.
3rd person over the should fixed perspective is just 1st person with extra steps. To me that is the same genre. It gets interesting, if you have free camera or at least free rotation.