I wonder how that would play out in today's Congress. There is technically no country to declare war on, unless you can declare war on yourself, so they would have to first redefine the United States as two parts. After that is passed, they could declare war on the opposing half. I guess I'm kind of seeing how the Chinese governments got themselves tied up in knots with their continuity of state and all.
I'd imagine that you can declare war on a specific group (that does not have to be defined as a state) like the "confederate states of america" without it having to be a nebulous concept like "terror". But I don't know enough to say for certain.