The good old question :)
For me it lies in the wording:
1. massively: so I would expect at least several hundred players simultaneously on one server. Session based games excluded, as mostly I have only communication access to like 32 or 64 players at the same time.
2. multiplayer: the game features real multiplayer modes. This means it should be possible to either meet other players in game or do any kind of interaction with them. This also includes browser based strategy games, often visualized with static images and tables, but finally I am able to attack any other player on my realm.
3. Online: Online is misleading here. The game should not only have an online mode but more or less should offer a virtual world with persistent game elements. Not the whole realm has to offer persistency but certain core game elements should offer respective persistency.

Not yet played Borderlands, but always thought it is kind of fallout gameplay with session based multiplayer modes. If this is so, it wouldn't count as mmo following my definition.

Thorsten