I think C.A. is right about the division, Elfwine. The Theodosian Code refers only to "the Empire" in singular, I think. The idea was a unified Empire that just happened to be run by two courts for reasons of administrative convenience, rather than two separate allied states.
I cede the point on his intentions.
But I'd still say that de facto we see a division that did mean that the term "Western Roman Empire" mean something.