RPGs in the sense of the genre (the italics make it important :3) are usually defined as games in which you impose a variant of yourself upon the game world, acting out whichever role you so happen to choose.
And whilst you are technically right, all other games do not allow you the freedom of choice to play whichever role you so wish, which is what makes an RPG an RPG.
Take Fallout 3 for example, the game gives you a story and an idea of what you're meant to do to complete the game, aside from that, it gives you no barriers, you're free to project yourself into the world and carry out whatever role you like, from the humble trader to the town drunkard, from the bounty hunter to the merciless killer, from the tech hungry scavenger looking out for his or her self to the Samaritan of the wastes, willing to give up their last bottle of purified water for the good of another.
Then you compare that to the majority of other games, the FPS games, TPS, RTS, action adventure and so on, these games have a strict, very linear storyline in which you, yes, act out the role of the main character, but the difference is, you have NO freedom, you're forced into that one role and you cannot deviate from the set path.
It's like the age old argument of "Sticking a J in front of RPG does not make the game an RPG any longer."
And this is largely true, whilst you're still playing a set role of a character, that's just it, you're only playing a set role, you no longer have the vast freedom of choice that the traditional RPG genre always affords.
That is what makes RPGs true role playing games, and other games less so.
