I went to a driving class and saw several different "minis", but to me, only some looked like horses. mini1.jpg mini2.jpg mini3.jpg
Edited to add: it was a great show. Congrats to all are in order.
I found this on the web:
THE AMERICAN MINIATURE HORSE OFFICIAL STANDARD OF PERFECTION
General Impression: A small, sound, well-balanced horse, possessing the correct conformation characteristics required of most breeds. Refinement and femininity in the mare. Boldness and masculinity in the stallion. The general impression should be one of symmetry, strength, agility and alertness. Since the breed objective is the smallest possible perfect horse, preference in judging shall be given the smaller horse, other characteristics being approximately equal.
Size: Must measure not more than 34 inches at the withers, at the last hairs of the mane for AMHA or 38" for AMHR. HEAD: In proportion to length of neck and body. Broad forehead with large prominent eyes, set wide apart. Comparatively short distance between eyes and muzzle. Profile straight or slightly concave below the eyes. Large nostrils. Clean, refined muzzle. Even bite.
Ears: Medium in size. Pointed. Carried alertly, with tips curving slightly inward.
Throat Latch: Clean and well defined, allowing ample flexion at the poll.
Legs: Set straight and parallel when viewed from front or back. Straight, true and squarely set, when viewed from the side with hooves pointing directly ahead. Pasterns sloping about 45 degrees and blending smoothly, with no change of angle, from the hooves to the ground. Hooves to be round and compact, trimmed as short as practicable for an unshod horse. Smooth, fluid gait in motion.
Color: Any color or marking patterns, and any eye color, is equally acceptable. The hair should be lustrous and silky.
Show Disqualifications: Excess height. Monorchidism in Senior Stallions. Any unsoundness or inheritable deformity. If in doubt, the show Judge may request the opinion of the show Veterinarian. Non-disfiguring blemishes not associated witlh unsoundness, or injuries which are temporary, should not be penalized unless they impair the general appearance and or action of the horse
So, it seems pretty subjective to me, other than as dirtymartini suggests a foal of two registered minis should be a registered mini, but may not be show material.