The term CAR comes from Colt Automatic Rifle. Regardless of what AR stood for under Armalite, when Colt bought the rights, they chnged the name. To build upon what name recognition the weapon had, they added the "C" and changed a word or two. It was strictly a marketing move by Colt. The entire AR-15 system was refered to as the "CAR-15" system and sold under that moniker. This included the rifle, heavy barrel squad weapon, carbine, sub-machine gun, survival rifle, everything was a "CAR-15" because of marketing.
When the Army started playing with the submachinegun version as the XM-177 series, the civillian market name managed to transition over and became one of the gun's nicknames along with "shorty" et al. As no one really bought any of the other versions, Colt eventually dropped most and the CAR-15 name applying to other than the one that was used (the Shorty) was forgotten. The CAR-15 name was forever linked to the shorter weapon from then on, even though a whole line of "CAR-15's" had been developed.
As the "M16" name became world renown, Colt renamed the remaining guns in the system the "M16A1 weapons system" and included a rifle, carbine (basically an XM-177 with a light weight 14.5" barrel and standard FS), and a heavy barrel squad support version. Again, this was a marketing move to take advantage of the name recognition of "M16A1". I have a Colt catalog of the "M16A1 system".
That's the true story of where "CAR" came from. As for what AR stood for, I'd have to think there was "Armalite" in there somewhere when it was in Fairchild, but it's not the same company as today's (which is a perfect example of name recognition at work).
Ross