For what it's worth, I have a copy of an old Kynoch catalogue from the '30's here and it lists .275 Rigby and 7mm Mauser separately, and with slightly different ballistics.
At the top is the ".275 Rimless (Rigby)", in both 140 and 173 gr, loaded to 18 tpi pressure. Immediately below it is the "7mm (.276) Mauser (Spanish)", same bullet weights but loaded to 16.5 tpi for the 173 gr. and 18.5 tpi for the 140 gr.
Bell used German DWM ammunition in his Rigby, and used all three names in his books, 7mm, .276 and .275.