Inu no Taisho
'Inu no taishou' is how he is referred to in the manga. It translates to 'leader of the dogs' and is a title, not a name.

Inutaisho
He's most commonly called this in the fandom, and Inutaishou could be considered his fandom name. The first documented use of the name Inutaishou was in the fanfic 'Under the Dog Star' by Catwho. It's essentially a shorthand way of saying Inu no taishou, and it sounds more like a name than a title. 'World to Come' uses the name Inutaishou for convenience, but recognizes he does not actually have a name.

Touga
Touga was the name used to refer to Inu no taishou for story boards and such during the making of the third Inuyasha movie. They never actually name him in the movie, however, and in the sound track he is listed as 'Chichi' (father), so it's safe to say it is not his actual name. Touga translates to 'fighting fang'.

Inu-Papa
Well, it's not actually a name. A nickname that the cast of the third movie used to refer to him, and the fandom has adopted it.