Unfortunately, some people believe that all pintos (horses with colored markings) are Paints and that all Paints are pintos...which is just simply wrong (though I am guilty of calling most colored horses "paints" regardless of whether they are APHA or not, just because that's what I learned growing up and it's hard to change the habit LOL).
Like Chiilaa said, pinto is the actual coloring of the horse. Any
horse of any
breed that displays white markings beyond their legs and face (depending on how far they extend) is a pinto.
Spotted saddle horse
American Paint Horse:
And the inbred little grade filly belonging to the neighbor
All those horses are "Pintos".
However, only a horse that is registered by the APHA can correctly be called a "Paint", whether they display pinto markings or not. Unfortunately, some people cannot see the difference...or choose not to.
Simply put, a Breeding Stock Paint, or non-pinto horse registered by the APHA (like your mare), can be any color including dun and roan and cream dilutions.
As for finding out her zygosity and all the genes that she's carrying, you'd have to pull a hair sample and send it off for testing.