What are you referring to as a "mid-sized"?
The American Miniature Jersey Cattle Association registers Miniature Jerseys that are up to 46" at the hip (both parents must also be registered).
The International Miniature Cattle Breeders Society separates into standard, mid-sized, and mini, and will register animals with no other requirement other than height -- registrations through the IMCBS are NOT recognized by AMJCA.
AMJCA animals can trace their ancestry back to the Jersey importations of the 1850's, and there are less than 100 registered full-blood animals in the US. They managed to avoid the "big is better" movement, so basically they are the "original" Jerseys -- virtually unchanged from the Isle Jerseys. And most of them are polled.
Ann B