Here’s what Hackley, Woodin, and Scranton, Vol. III says:
"In early 1962, Remington submitted documentation to the Sporting Arms and Ammunition Manufacturers’ Institute (SAAMI) to register the .223 cartridge and chamber dimensions, which was then entitled “.223 REM.” This was approved by SAAMI on May 17, 1962.
In May 1962, the U.S. Air Force received approval to procure limited quantities of the AR-15 rifle and 8,500,000 rounds of .223 ball ammunition for testing, training, and unconventional warfare use. On 25 January 1963, the Remington .223 Ball - 55 gr. M.C. cartridge was adopted by the Air Force as Cartridge, 5.64mm Ball (emphasis added), MLU-26/P (FA Dwg. D10523311, 1 May 1963). Later this nomenclature would be changed to reflect the caliber as 5.56mm (ASD Card 219-26, T.O. 11W3-5-5-1 Fig. 5-1). Despite the large procurement of .223 ammunition during this period, no box label has been examined having the MLU-26/P model number, and it is assumed the rounds contracted for by the Air Force all used the Remington .223 - 55 gr. M.C. designation."
Several headstamps of the period are illustrated, and some have REM-UMC, but those also include 223, not the 63 date, which, as Ray said is seen with R A, not REM-UMC.