No, I think it's:
Halo 1 multiplayer w/ Halo 1 maps
Halo 2 multiplayer w/ Halo 2 maps
Halo 3 multiplayer w/ Halo 3 maps
Halo 4 multiplayer w/ Halo 4 maps + 6 remade Halo 2 maps
Am I right or way wrong?
Each game runs on it's own engine with it's own multiplayer.
I'm not too sure about the 6 remade maps.
But what you said about each game with it's own multiplayer is true.