When Colt bought the rights to the AR-15 from Armalite, they changed the name to CAR-15 for Colt Automatic Rifle-15 (as opposed to Armalite Rifle-15) to capitalize on the marketing value of their own name and the name of the AR-15. It's all smoke and mirrors marketing.
They developed several versions and marketed them all as the CAR-15 system. Everything from SMG, Carbine, Survival rifle, rifle (the M-16, which they referred to as CAR-15 rifle) and heavy barreled support guns. They were technically all CAR-15's when offered by Colt. The Government tried them all, but only became interested in the SMG which became the XM-177 series of weapons. It was sometimes referred to by it's commercial name CAR-15, even though "CAR-15" was a whole system and not just a single gun. The name stuck, and the XM-177 series was frequently known as the CAR-15. Many thought that it meant carbine or whatever, but it didn't.
Marketing being what it is, Colt changed the name of the CAR-15 system to the "M-16 system" and offered a carbine, rifle, and HBAR under that moniker. The M-16 name had become worth more than "Colt" had. The carbine offering (a refined XM-177) was further developed into the current M4 carbine.
Marketing also being what it is, most aftermarket companies offered their collapsing stock versions and these were referred to generically as CAR-15 by most of the AR world. In the end the term AR-15 would become the generic term for the rifle version with 20" barrel. The term CAR-15 would become generic for the collapsing stock carbine version of the same.
So now you will find most rifles referred to as AR's or AR-15's even if they are officially a different name and maker. You'll also find the Carbines referred to as CAR-15, also regardless of the model or maker. The term CAR-15 has lost a lot of usage as we can now no longer have collapsing stocks, and more post-Vietnam folks enter the AR world.
Ross