The truth is yes and no,the Staffy or staffordshire bull terrier is the fore runner to the American Staffordshire Terrier/ APBT both of which were bred from the original Staffies.
Now the Staffy was considered a "purebred" until 1935 when they wrote the standard for the breed.
That said,Staffies had been bred for over 100 years before the standard was wrote,this obviously was done because of the fact that dogfighting had become illegal and they needed a reason to continue to breed the dogs. So "purebred" means dog shows and a reason was made!
Long before this though,in the late 1800 many english and Irish immigrants came to America and brought with them their fighting dogs,mostly of the Staffy type,but many other breeds too.Soon these dog became popular and bred to slightly different sizes,the American dogs got slightly larger than what was found back home in England.
So over the last 100 years or so the two dogs started to differ greatly and were named differently depending on who was registering the dogs.The AKC refuses to recognize the APBT as a breed but recognizes the AmStaf,both are basically the same dog just bred to a slightly different standard and purpose.
Ok so now I wrote a book......LOL but the answer is still yes and no!!