You're pretty close. Those rounds are measured in millimeters instead of caliber, so they should be 7.62mm (really 7.62x39mm or 7.62x51mm) and 5.56mm. The equivalent in calibers is .30 and .223. Notably, .223 is also an actual round and some rifles can fire either .223 or 5.56 (iirc, the cartridges are the same dimensions, but the pressure ratings are different).
AR-15's are typically chambered in 5.56mm. An AR style rifle chambered in 7.62x51mm would be an AR-10. Although you can get AR platform rifles chambered in pretty much whatever you want, going all the way up to .50 BMG. It's pretty uncommon to see anything like that, though.
