Estelle,
A mini at this point in time is purely a HEIGHT breed. It is not yet genetic. A miniature horse according to the American Miniature Horse Association is any horse that fits their conformation standards and is 34 inches or under at the last mane hair of the withers. AMHA will recognize all horses/ponies no matter what their genetic background if they fit their registration requirements, however, the Hardshipping is spendy......... According to the American Miniature Horse Registry it can be any horse that measures 38 inches or under with the same standards as AMHA, but they only recognize horses that hold registration papers with AMHR, AMHA, or ASPC (or have parents that do.)
Hope this makes sense. I probably should have typed this out in separate paragraphs so it was easier to read.
MA