People seem to have decided that closed rpgs are somehow not rpgs, despite them actually being the more common form of rpg. Any game that has you assume control of a character that has a name, a role in the narrative of the game, either pre-made or not, and contains some form of character development (leveling), is an rpg.
A closed rpg is one where you basically have no choices, your character is predetermined and the story is predetermined, you are along for the ride, but not in control. Most of the Final Fantasy games are all closed rpgs.
Behold, a rpg!