difference between original xbox 360 and the rerelease?

Wanting to buy my partner an original xbox 360 for our anniversary bc they still have their old hard drive from when they were a kid but I do remember them mentioning that the storage system is different in the original vs the later rerelease so I'd love some help knowing what to look for in the online listings so I know I'm buying the proper one - they had an original and not a rerelease to clarify.