Should the barn be responsible for my horse's injury?

Depends on how far you want to push it. Check your contract if it has a not responsible for accidents clause you may have to argue it. However if they openly admitted that it was left out you have leverage. Its a double edged sword. Be prepared to be made to feel incredibly unwelcome if you push the issue because I cant for see them paying and saying sorry very lightly. Hope everything works out for you guys.