Does anyone have a good pic of this?

Close as I can get from here.

This might be a little better.

Why was it designated sub-machine gun? If my understanding is correct, sub-machine guns are pistol caliber.

Many terms end up being imprecisely used.  From a technical perspective, you are correct.  But from the military's perspective, remember that these carbines were probably replacing M3 grease guns in many applications.  The XM177E2 was referred to as an SMG as well.

Federal Item Identification Guides (FIIG) are used when assigned nomenclature in any of the designation systems and to commercial items.  The FIIG for Guns and Related Components, FIIG T102, has the following definitions:

Submachine Gun: A weapon fired from the shoulder or hip, with a barrel 8-1/2 to 11 inches long, using small arms ammunition and capable of semi or full automatic fire.

Carbine, 5.56mm: A small arms weapon with a short barrel and stock, magazine-fed, gas operated, 3-round burst or semiautomatic, which is usually fired from the shoulder.

There is no basic definition of Carbine in the FIIG.  Carbine, .30 Caliber and Carbine, 5.56mm are defined separately.  Really, whenever a nomenclature has to be assigned, it appears that generally someone opens up the Cataloging Handbook, H6 and finds an appropriate term, then looks up the relevant FIIG to determine the Federally Approved Item Name.  Terms can be added to FIIGs as well, but I assume this is only done when no term is at all close to the item being classified (The M231 is also classified as an SMG, and clearly fits the definition in the FIIG).

On a separate note, its GAU-5A/A, not GAU-5/A/A.

GAU stands for Gun Aircraft Unit  but they were not installed on aircraft in the Air Force that's why the designation was changed to GUU which stands for  Guns miscellaneous personal

Thanks for the pic AFSOC. What does it say under SMG GAU5/A and above the s/n? It looks like another number of some sort but it could be a shadow of some kind.

5.56mm

Wow I'm a freakin retard. Thanks man. You'd think I'd remember, I was issued one of these at one time and that, coupled with common sense should have told me.