There are 2 very different cartridge groups which are often confused:

1. the 9.3x57R/360 is a very old cartridge (say 1880s), descending from the British .360 Express 2 1/4, conical or tapered case, dimensions very similar but not identical to the Bitish case.
This was the time before cartridge dimensions were standardized, so there were different versions with respect to case shape, bullet diameter, and loads (black and nitro).
There never was a 9x57R/360, but versions copying British dimensions were closer to 9 mm than 9.3 mm (but to my knowledge always called "9.3" in Germany).
The 9.3x57R/360 must have been quite popular at some time, but was then quickly superseded by the longer 9.3x72R.
Cases can be formed from the 9.3x72R. Base diameter is about .430.
There was an article by Ross Seyfried in Handloader, 2003, and another one by Helmut Eller in Deutsches Waffen-Journal, 1994. I can send you these as .pdf files.
I also have some own experience and loading data.

2. the Mauser family: bottleneck cases, base diameter about .470.
there was a 9x57 (rimless), a 9x57R (rimmed), a 9.3x57 (rimless), but to my knowledge never a rimmed 9.3x57R.

Fuhrmann