Lloyd is correct here. Bully means basically anything with the Olde English Bulldog or English Bulldog in the mix. The term has be watered down to mean anything with any bulldog blood such as American Bulldog and all the other "bullys".  A quick search on here will show several bully types in many countries.  Now when it comes to the Bandog - technically there is not supposed to be any Bull in the mix at all - at least not directly - The American version of the Bandogge is supposed to be APBT and Neo so no reall Bull unless we stretch back to saying the APBT has Bull and so does the Neo.  Anyway here is a snipped from our Bandogge profile. so, no, the Bully is not a Bandog.

"The widely accepted recipe for an American Bandog is usually 50% American Pit Bull Terrier and 50% Neapolitan Mastiff, although some breeders use other mastiffs as well, like the English Mastiffs, Bullmastiffs, Cane Corsos, Rottweilers, Japanese Tosas, American Bulldogs, Brazilian Filas and so on, as well as English Bullterriers, American Staffordhires, Irish Staffies and others for the bull-n-terrier part of the equation. Interestingly enough, there are not many known bandog programs involving the use of Molossers such as the Central Asian Shepherd, Irish Wolfhound, Spanish Mastiff or any of the old terrier and hound breeds, all of which would be historically more correct than the modern "American cocktail", which relies on breeds which didn't even exist at the time bandogges "ruled" Europe. There is a number of modern Bandog breeds worldwide, such as the Czech Bandog, Australian Bandog, German Bandogge, Brazilian Bullmastiff, Ambullneo Mastiff and others, but most fanciers still associate the Bandogge name with the Swinford-concocted recipe, where the father should be a game-bred bull-n-terrier breed and the mother should be a large mastiff-type dog" Read more...