There was never a rimless .303 case, at least not in England! But there is a 6.5X53R rimmed Dutch Mannlicher case that is about the same as the rimless 6.5X54 Mannlicher-Schoenauer case. A number of British rifles were built using the rimmed version, and both of these rounds would look a lot like a 6.5X55 Swedish Mauser case, except they are smaller around at the head.

Keep yore powder dry!!