The original name isn't pitbull, though. The original name is bull and terrier (mixing a bulldog and a terrier). It then became bull terrier. The pit part came much later. You only referenced the pit part, which makes your post seem disingenuous.
They did not get their names from ratting. They were called RATTERS if they were trained to fight rats. Pit fighting, literally fights in a pit, happened with other dogs.
Additionally, during rat pitting, there was another dog that was in there that would compete against the first dog, and at the end, it often ended in dog fighting that was not immediately stopped (conveniently). It was just another front for dog fighting. /: