It was originally referred to as .222 Special, which was the version produced for Gene Stoner from the specs for the last version of Aberdeen Proving Ground's SCHV experimental ammo. Since there was already .222 Rem, and there were plans for .222 Rem Mag (intended to be a civilian version of the SCHV) they later changed the name to .223 Rem before commercial and military sales of the AR15 began. BTW, the .223 was not a scaled down or necked down anything - it was based on the .222 Rem, but with increased case capacity and a heavier and specifically different projectile. It very nearly never saw the light of day thanks to one Dr. Fred Carten and his championing of the 7.62x51 as the only battle cartridge needed, despite the fact that the SCHV was intended by its original designers as a replacement for .30 carbine and all subguns, not as a main battle rifle cartridge.
.222 Rem Mag quickly became obsolete when the military adopted .223, and is no longer available.