I know what the Dominate was, but it was the main Roman Empire, still ruled from Rome, not a continuation.
I don't see any difference between those things. A "continuation" means the "same thing continuing"? A continuation is not a sequel.
I'd put both the Dominate and the Byzantine Empire as the same thing. I actually hate the name "Byzantine Empire" because of how anachronistic the name is. It WAS the Roman Empire. If you need to distinguish it, just call it the "Eastern Roman Empire".
I also put the Papal States down as, while not as clear cut as the other two, it was a direct continuation of the state religious apparatus of the Western Empire.