The truth is the relationships between the breeds can be charted chronologically, but the rest of it is murky enough that it sometimes depends on who you’re talking to. In particular the two American breeds are so similar that calling them two breeds is questioned by some experts; much confusion exists on the issue.
My understanding of it, after a lot of reading (mostly Internet) is as follows:
Staffordshire Bull Terriers (“Staffies”): the smaller British dogs.
American Pit Bull Terriers (APBTs): the formal modern breed name for the Americanized dogs derived from Staffies over the past two hundred years or so. These are what most people are thinking of when they say “pit bulls,” although confused and emotional “thinking” is common in regard to pit bulls.
American Staffordshire Terrier (“AmStaffs”): these dogs are really so close to APBTs that many dogs carry dual registry as both “breeds.” My understanding is that AmStaffs aren’t really as much heavier than APBTs as the numbers you’ve got suggest; it’s more that size isn’t discouraged in the AKC show dogs. The most visible difference is that AmStaffs have to have black noses and APBTs can have red or black noses, and the AmStaff champion lines favor a slightly blockier look for the show ring. AmStaffs were “developed” (if indeed they’re separate enough from APBTs to use that term) last as a way to get APBTS into the show rings.
Again, it’s my sense from what I’ve read (as opposed to Revealed Truth) that the UKC was started in part by people upset that the AKC wouldn’t register APBTs (which supposedly was because fighting dogs were low class, in a way that dogs killing foxes and rats somehow was not, heh.)
It’s been argued that the “dogs bred for fighting” are a distinct breed, and a very carefully-managed one at that – supposedly the old-time “dogmen” carefully tracked genetics and worked to make the dogs as athletic and determined as possible, although for entirely the wrong reason, of course. That might be true, or it might be salesmanship or braggadocio for all I know.
All three breeds (Staffies, APBTs, and AmStaffs) can be legitimately considered “pit bulls.”
Lots of much larger dogs exist, as backyard breeders have been crossing them with bigger dogs to impress the sort of people who are impressed only by size. I think mastiff is a common outcross. These larger dogs are mutts, but the people selling them call them pit bulls.
There is currently a movement afoot to designate the wide-body, big-headed mutants some people have been breeding as a fourth distinct breed under the umbrella category of pit bulls, and call these dogs “American Bullies.” Google American Bully pics to see what I mean; these dogs are much bulkier and usually larger than the athletic APBT. Fans of the original APBT have been known to call AmBullies “hippos,” perhaps you can see why.
Whether such a designation will stick and become formalized is anybody’s guess, like deciding what slang will stand the test of time.