Edit: There are people who think that aggression towards small animals = aggression towards dogs = aggression towards people. They are all distinctly separate things. A dog that attacks a cat is not a dog who will necessarily attack a dog and is not dog who will necessarily attack a person.
The problem is that people bunch dogs into categories of either "friendly" or "not friendly". People who think pits should be held to different standards because one attacks a cat are wildly ignorant. On the flip side, pit owners who think they can leave their dogs unattended around small animals because their dog is great with people are wildly ignorant as well as being irresponsible owners --- and terrible for the breed. It sounds like you fall into the former group, while your friend's mom falls into the latter group, no?