I guess these couple of paragraphs taken from above provide the best enlightenment I've seen of the old argument, is the .275 Rigby cartridge a different cartridge than the 7x57. It would seem not despite what may be on the Rigby cartridge packet and head-stamp.
No different than say Australian made AR2208 powder repackaged and sold as Hodgdon Varget powder. It is still AR2208 powder. The .275 Rigby is still the 7x57 cartridge.

John Rigby & Co. had a business arrangement with Mauser of Germany in the late 19th century, importing their fantastic Model 98 actions and giving them British stocks and appointments. Rigby offered the 7x57mm Mauser as one of the chambering for their rifles—understandably, it was and is a fantastic hunting cartridge—but decided to rename the cartridge to make it more palatable to the British market. This wasn’t uncommon; the .404 Jeffery was known as the 10.75x73mm on the European continent.
Hornady-275-Rigby
The .275 Rigby is simply the 7×57 Mauser using an alias. Photo: Massaro Media Group.

The .275 Rigby is, was and ever shall be nothing more and nothing less than the 7x57mm Mauser, and that’s certainly not a bad thing. The classic design uses the full gamut of the 7mm bullets, with the 175-grain bullets having made the reputation for the cartridge. But anyone interested in the .275 Rigby should be aware of the .275 Rigby HV chambering. The lead of the chamber has been revised to optimize performance with the 140-grain spitzer bullet; the HV chamber won’t accept the 175-grain bullets, and only a handful of the 150-, 156- and 160-grain bullets will work in the HV chamber.