I haven't read this entire thread so sorry if I am repeating what you have already been told. A paint horse is a breed. They have lineage that is traceable and all registered horses. Yes, the APHA horse can have AQHA and TB lines in it but can not be bred to any other breed and still be registered a paint.
Pinto is a color registry. You can have a horse that is registered pinto and not be a paint but you can have a paint and have it registered APHA and PtHa.
I found this post and your first post to be confusing. A paint is not a quarter horse if it were it would be AQHA NOT APHA.
I have a feeling I worded it completely wrong. Paint is a breed pinto is not. Horses of stock body types and specific bloodlines (AQHA, thoroughbred, ect...) It can be registered APHA only if one or both parents are registered under APHA, AQHA or the Jockey club.
If the horse exhibits pinto markings but is NOT of stock breeds or bloodlines ( saddlebred, arabian ect. Ect.) it can only be registered pinto.