Submachine guns are defined as weapons featuring select fire and pistol caliber chamberings. They may or may not be small. Some have been quite large. Thus, submachine gun wouldn't be a proper description for a short long gun firing pistol cartridges in some cases - but it might be. Not that that would stop anyone from misusing the term, which brings us right along to...
Some things to ponder:
My Mechtech Glock upper is referred to as a carbine when installed. It is not a shorter version of an existing rifle. (It comes in one size only.) It fires a pistol cartridge. (.45 ACP.)
Dragoon and cavalry carbines of the 19th century were shorter versions of an existing rifles. They fired rifle cartridges, for the most part.
The M16 is referred to as a carbine in some military publications. It is not a shorter version of a parent rifle. (It is the parent rifle.) It fires a rifle cartridge. One could make an argument that it means "short rifle" in this case, not "shorter", but that wouldn't make sense; it's not a particularly short rifle and there are shorter rifles in existance that are not referred to as carbines.
We have three different meanings of carbine above. Which one is correct? All of them, depending on who you ask.
I can understand why Dave is confused. "Carbine" gets tossed around quite a bit. As I said, it's pretty much a marketing term at this point as far as I'm concerned, unless applied to 19th century cavalry and artillery rifles. I agree that that was the first use of the term. I disagree that it's the only way the term is used today.