strictly speaking the animal OWNER is liable for any damage, but in the event of him/her not having insurance, then the property owner may be joined in and the case made against him/her on the grounds that fencing was not of proper standard.
Your House insurance may not cover you in this instance. it normally covers house and garden. If the land horse is on, is grazing then it is not garden and you may actually need extra cover. I assume FBD would be your best bet, especially if horse owner had his/her cover with them.