The "changing gene" is most likely separate from Piebald if there truly is piebaldism in bettas. If it's truly piebald the pattern won't change, ever. It will always be colorless in the same spots. And it may look white or pink, but it's not, it is colorless, different than having just a white spot.
But in Humans and snakes and horses and dogs, the trait is recessive. But you have professional breeding and AKC -type breeding standards that have existed for hundreds of years so in many of the breeds that have piebaldism, it's a trademark of that breed, and you'd be hard pressed to find any in that breed that don't at least carry the gene for the trait, even if they don't display the trait (as in they are solid).
Again people in the Betta selling profession throw terms around that their fish may not exactly display. Everything that gets called marble is potentially 5 or more different genetic conditions. Just because you've seen it called pied or piebald, doesn't mean it has the genetic condition for which it's named. It may just be a fish with a white spot on it.
My point is, there is a True, real condition called piebald that is well documented, and it may or may not be present in bettas. If it's different than the way i describe it, it is most likely not truly piebald. Someone just decided to call it that because it looks similar to it.