Is the American Staffordshire Terrier a cross between Bulldog and game terrier originally called Pit Dog not in actual fact Bull?

No, the American Staffordshire Terrier is not a cross between a Bulldog and a game terrier. While this breed was originally called the Pit Dog, it is not actually related to the Bull breed. The American Staffordshire Terrier is a member of the Bull Terrier family and was developed in the United States from the Staffordshire Bull Terrier, which was brought over from England.