They are called Australian Miniature Ponies. Maybe if you Google American Miniature Ponies? I believe the sire of that second stallion I showed you is American. Here is a site for the AMPS.
If I got a shetland it would definitely be a true to standard one, not an Americanised one.
Incidentally, I have the height before in inches because my friend isn't talking metric yet. LOL. Should be by now but I still convert things myself too.