I think the closest definition by far would be H@H@'s definition.

To me its more of a Mulitplayer (Not Massive) Online RPG.

Since its only a SINGLE controllable unit with character progression. Same like diablo, neverwinter nights, dungeon siege etc.

Although Warcarft 3 is an RTS game the DotA custom map isn't. You should try other custom maps of Warcraft 3 too like footies, loap, tower defense etc. Its all in the same league (ie just mere custom maps).

I think its close to the PVP aspect of guild wars (though you don't level up that much in 1 match of guild war).

But basically its just another custom map of Warcraft 3.