You realize the fact that they have different names, whether or not they're officially inclusive of "north" or "South" means that there is no officially unified "Korea".
Because of this, when discussing "Korea" it only makes sense to specify WHICH Korea you're talking about.
You know, because there's 2 Koreas.
It doesn't matter if you call them "North" or "South", or if you refer to them by the official names. They simply aren't one, singular, nation.