You are right. The goldsmiths likely started fractional reserve banking. The first paper money backed by gold recipts for storage deposits of gold. Then they started trading the recipts as money good. Then the gold storage places figured out "hey, not everyone asks for their gold at the same time. We can just print up more recipts and treat them the same as gold".

Works ok till a run on the bank or the bankers get greedy and issue way too many recipts for the assets they have. Hey, doesn't that sound a little like today? Well at least the money is not backed by gold so no problem there.