Both ABCN and ABPN "count".
To be clear: ABPP is a completely separate, free standing psychological body. So is ABPN. Both organizations are affiliated with division 40 of the APA. ABPP is not part of the APA. Many very big names in neuropsych are boarded in abpn, so I think it would be difficult to say that the method by which Ralph Reitan, Cecil Reynolds, etc got their boarding doesn't count. Both boards are recognized by states and courts. The big difference is the Houston Conference Guidelines, which were not created until like 2004. Many senior ABPPers did not have training that met HCGs, which begs the questions: "do they believe this training model is necessary. And if so, are the senior members admitting that their own training is inadequate?". The significant difference is that ABPP was created in the VA system because a wise person realized that there was a pay step for providers who were board certified. As a result ABPP is recognized in the VA, whereas ABPN, which was created outside the system, is not.
ABMP is a bit of a different story, but you are missing some ideas. The practice of medical psychology is recognized by certain states, the DEA, and the APA. The APA has some suggestions moving to requirements for educational standards for MP, just like APA approved clinical programs. The APA also have a standardized test for medical psychology, passing of which is required by all states before one can practice as an MP. The states get to set the ultimate requirements for the practice of medical psychology. It's a recognized specialty by state law, federal law, etc. But the practice of medical psychology is not recognized in the VA.
The difference is that, aside from a few states, any psychologist can practice neuropsychology with really no extra training. Boarding is one method to demonstrate to patients and other clinicians that one has the extra training. However, boarding has basically no legal standing outside a few states. In medical psychology, demonstrating compliance with training and educational standards is required before one can practice.