The Pet Agreement (TXR 2004) pertains to all pets that the tenant keeps on the property, including in the yard. This means that even a pet that mostly or always lives outside must be listed in the agreement. Pet refers to any animal, whether mammal, reptile, bird, fish, rodent, or insect (except that an assistance animal is not a pet).
I think it should. If it’s about destruction to property, it happens outside, too. There is still the liability issue with people even if the pet is in a fenced area.