Yes, the horse owner/rider could be held liable ...I am not up on civil law
that said, someone could have let it out of the field, someone else could have been in charge ... and at the end of the day it is an animal with a mind of its own and most of them have a habit of doing exactly what they damn well please

A run route or a group of runners run the risk in the country of meeting whatever the countryside can throw at them from herds of cows/sheep, the local hunt or a fleet of combines, you deal with what you have to the best of your advantage or safety ... IMO