I've never dealt with .360 or 9.3 myself, but am familiar with and have reloaded for the 9.5X57 MS.

From what I've just seen poking around on the 'net, there seem to be several 9.3's.

You 'threw me' for a bit with "9X57 MS". The 9mm Mannlicher Schönauer cartridge was 9X56 (for the M1905), not x57. The 'X57' MS was the 9.5X57 (.375 Nitro Express Rimless 2.25") for the M1910 which is at times referred to as 9.5X56 or 9.5X56.7 just to confuse things further.

Here are some Mauser specs, including the 9X57:

A link to previous discussion among members regarding the 9X57 Mauser, 9X56 MS, 9.3X57:

NE Forum