I did it on the old mac, and - lo and behold - the new thing actually appeared for the first time, mounted, on my new computer. However, I added a folder named "MUSIC" with a song in it, tested that I could play it (I could) - and then the disc did not appear in finder again once reconnected with my new computer. This is very similar to what happened the first time it appeared. Also, I tried to partition again, and it just does not work! It seems to work once and never again...
Regarding the actual partitioning, I have chosen MS-DOS file system (FAT16) format, is that alright?
Any more ideas? I am very impressed with your lack of despair 😛