"What defines a shaving soap?"
I would say that for me it equals a bit of slip added with clay. I use white Kaolin, others use a bit of Bentonite. Also, perhaps a bit more generous lather.
"I wish I could get my round soaps to unmold this nicely."
You can I just use PVC. After made several batches they do come out a bit easier. Try freezing and if you have a bit of experience you can discount your water a bit.