Wikipedia states that a "Role Playing Game" is:
Players assume the roles of a character, or take control of one or more avatars, in a fictional setting. Actions taken within the game succeed or fail according to a formal system of rules and guidelines.
I think it's the best definition of role playing I've ever read.

In fact every game where you can assume role of one or more character is a RPG. This means that RPG are not only videogames like Final Fantasy or MMOPRG, but in the RPG genre you can find play-by-chat RPG and games played with sheets like D&D too.

Then you can distinguish between:
-RPG where the story is just written and it doesn't depend on your action(like the offline RPG)
-RPG where the only objective is to increase your pgs parameters to enjoy with friends(like MMORPG)
-RPG where the story is created by the actions of your pg and other pgs(like in play-by-chat)

Then you can also say that in classical meaning RPGs have a fantasy background, but now there are other kind of RPG from every genre.

Sincerly I think you can call RPG every game where you can join and have a specific role in the story.

My indecision is only about MMORPG that are called RPG but there isn't a story...so why they call them RPG? I know it's very difficult to create a story online in real time when people come and go continuosly but maybe a general story may be interesting in them too...