Just to perfectly clear. Can you use different parent cases to make an overbore 22 caliber cartridge? Sure. But the 22-250 is unique to the Savage parent case, hence 22, caliber and 250, based on the 250 Savage parent case.
As I said above, use whatever case you want but the moniker, 22-250, is only correct when referring to a cartridge based on the 250 Savage case.