Here's my definition. If it meets these conditions, it's a role-playing game:
- There is a fictional world.
- All the players agree with what is happening in it.
- 'Playing' means suggesting changes to the fictional world.
- Players and refs decide what moves you can make by deciding if it is sensible within the fiction.
- The primary audience are the players themselves.
Where it gets confusing with board games and such is that it's quite possible to add in any new elements you like, like boards and rulebooks and such, usually to manage #3 and #4. But those elements don't tell you if something is role-playing or not, since they are missing from lots of RPGs.
Rule #4 is the really interesting, one, though, and needs a bit of explanation...
In a board game, the moves you make are defined entirely by the written rules. Chess is a classic example — a knight moves two ranks and one file, or two files and one rank, and that's that. You can never say "Through a brilliant act of horsemanship, my knight moves three ranks forward."
In a role-playing game, the moves are only constrained by the fictional world everyone is imagining. In a role-playing game, you can say "Through a brilliant act of horsemanship, my knight leaps over the high fence" — and everyone around the table decides whether that's a reasonable thing to happen. They might do this through, say, a dice roll, but they could just do it by agreeing it's reasonable.
So this is the key thing to look for: is anyone making sensible decisions about what happens in a shared imagined world, doing something more than just applying the rulebook?
So to classify some games:
Non-RPG miniatures game, like Talisman — It doesn't matter what you say, or how silly a move is — if the rules allow it, you can do it.
RPG with miniatures, like D&D — Think about conversations between the party and NPCS. If a player says something like "I levitate above the ground while we talk" another player will say "Stop being silly", and that doesn't happen.
Pictionary doesn't have a fictional component that everyone agrees on.
Werewolf/Mafia is interesting. I think it might actually be a role-playing game, albeit short-lived and limited. (I assume we're talking about the party game, not Werewolf: The Apocalyse). Consider a live role-play where there is a trial -- it looks a hell of a lot like Werewolf. I can't think of a reason to say one is and one isn't.
Freeform LARP obeys all the rules. Improvised theater looks very similar, except it doesn't meet #5 (audience).
Live-action Role-play also obeys the rules.
Computer 'RPGs' aren't like tabletop RPGs since the possible moves are not negotiated between the players (#4).