Drawing from your example, if you have offspring from two double registered APHA/AQHA horses, and you choose not to register with AQHA, it does not make that horse any less quarter horse IMO. It does, however, make it an
unpapered quarter horse.
Can you tell me what characteristics are different between the AQHA and APHA breed standard? When you look at a horse, what conformational traits identify it as a SPB instead of a QH?
The Jockey Club is more than 100 years old and is the American registry for thoroughbreds, which was a breed developed in the 17th and 18th century. It does not, and never has, accepted quarter horses or paint horses in its registry who were not also 100% TB with Jockey Club registered parents. The fact that APHA recognizes some purebred TBs as paint horses is neither here nor there.