Well there is really no standard rule about naming cartridges, but here is the breakdown on the cartridges ou mentioned.
7.62X51mm the Europeans usually designate cartridges this way, but this on happens to be the military verson of the 308 Win. 7.62mm = .308" which is the bullet diameter, the second number, 51mm is the case length. The 30-06 would be 7.62X63mm.

22-250 is named such because it is a 22 caliber bullet sitting in a necked down 250 Savage case. The 7mm-08 Rem is a 7mm bullet in a necked down 308 Win case.

Some cartridges are named afer a person or the company that developed them, i.e. 454 Casull was invented by Dick Casull, the 270 Win was commercialized by the Winchester company, the 7mm-08 Rem was commercialized by the Remington company.

Some of the older cartridges take their names from the blackpower loads they used to use such as the 45-70 was a 45 caliber bullet over 70 grains of blackpowder, the 44-40 was a 44 caliber over 40 grains, etc.

Then you have wildcats where anything goes. Hope this helps some.