Did you measure correctly? From bottom of hoof to top of wither ? Not the back.
The rule of them in my area, is under 14.2 h it is a pony. I have a 14.1 h mustang and quarter that I call small horses :) . Head size does not make a horse. My fjord is small also even though she is considered a horse for the breed. Quarter horses have had the height limit removed. There is a diff between a shetland, welsh pony, and the small horse pony. The shetland and welsh are quite small :) I hope you are not confused more.
It basically depends on the breed. My draft crosses are 16 hands and larger. ANd by the pic shown yours could be a pony .
And Seriously, if you are happy with them what diff would it make ? I love all my horses even the fugly ones :)