fluffbug...I'll answer here since others may want to know.
The American Miniature Horse is a height breed. There are two registries: AMHA and AMHR. AMHA requires the horses to be 34" and under. The measurement is taken from the last mane hair, unlike big horses where it's taken at the withers. Minis can be small--even 28" and under. AMHR has two divisions. The height requirement for A is 34" and below, while the B is over 34" to 38".
There's now such thing as a miniature pony. Ponys are over 38" and under 14.2 hands I believe.
How's that? :D