I'll hand a pass to the UKC as I haven't done a whole lot of research onto their ethics and what they do or don't allow in their books. I'm not keen on how they'll allow a full UKC breeding registration on an AKC limited registration dog that isn't supposed to be bred, but.
Until recently, the ADBA allowed blatant mongrels into their 'pureblooded registry' as seen through allowing Razor's Edge and Gotti line dogs into their registry, when the breeders admit openly that their 'pure bred APBTs' are crossed with other breeds, including amstaffs and mollosers, to get the desired look. These dogs were not expunged from the same registry, and even allows that 175 pound obvious molloser-cross mutt 'Hulk' in its purebred lines as an american pitbull terrier.
Only as of March of THIS YEAR have they bothered to attempt to define the APBT, Amstaff, Working Pit Bulldog, and American Bulldog as different with breed standards and limitations, and finally will not be in theory allowing crossbreeds between any of them. While the ADBA may be moving in the direction of a reputable legitimate registry, right now it isn't due to its polluted lines and mis-registration. Anyone who hands you papers from ADBA can be selling you a mongrel at pureblood prices.
Edit: Puppypix!! I love his little white feet.
A registry is third party that keeps a record of the genealogy of the animal in question.
What I consider disreputable registries is those who register dogs as breed X simply based on the owners say-so, or the opinion of a vet or sometimes some photographs.
AKC limited registration.
In my experience many breeders will offer limited registration and full registration for the same animal at a higher cost. It's not like the animal is given limited registration after a neutral third party decides the animal does not meet certain goals.
Second, UKC is about working dogs. An AKC animal may have an undesirable tail length or coat color for the show ring determined at birth and given limited registration, then turn out to be a fantastic hunter. Regardless, the exact ancestry of the animal is known going back many generations.
Regarding ADBA
You say mongrel like it is a bad word. It isn't. The APBT like the Bull Terrier and Staffordshire Terrier comes from a 'type' of fighting bull-and-terrier dogs, so by definition these all trace to 'mongrel' roots. This is true of most breeds.
When a group of humans got together and declared there was to be a breed called the APBT, then there was the breed APBT, and it was formed form the TYPE of dogs they were looking for. There were dogs that were registered as Staffordshire Terrier that were let in then. There were 'mongrels' that were let in then too. Today, because of breed bans a lot of APBTs were re-registered as American Staffordshire Terriers, so bringing them back in to the APBT registry to me only makes sense. As far as Razor's Edge and Gotti, yes these lines started with unpapered pit-and-bull types, plus some papered Staffordshires, and had some bandog thrown in for size, and may well have included some animals that were known to be just 1/2 or 3/4ths pit. What they continued to have was a TYPE, and just like in the founding of the APBT breed, it makes sense to include that TYPE.
What may seem odd is that people aren't used to OPEN registries. Closed registries mean they only take animals that are offspring of animals already on the registry (or recognized registries). OPEN registry means you have the ability to take in animals that aren't registered if they meet the TYPE.
Indeed these 'alien' animals may be closer to the desired goal animal that the majority of 'registered purebred' animals.
It used to be fairly common for breeds to get an infusion of outside blood, especially when the breed hailed from somewhere other than the UK or USA. For example, the Siberian Husky breed has had 'siberian type' dogs from the Siberian area imported and included in the breed.
And finally, re-opening registries and bringing in either unregistered dogs of the same 'type' or bringing in dogs registered as a different but similar breed is one of the few way the breeds we have recognized today will ever have a chance of overcoming their genetic bottleneck. This should not be a problem as long is it is documented and everyone knows what is going on.
In fact, totally closed registries are NOT the norm and is common practice in most other purebred animal species. Generally they have the stance of 'its purebred' as long as it is of type and has documentation to show it is 7/8ths pure.