Personal liability pays others. It is not meant for claims within the family or for property the insured is looking after. Three standard features show the pattern.
Injury to an insured is excluded. A relative who lives in the named insured's household is an insured, so if Ben's mother, who lives with him, trips on his rug and claims against him, personal liability does not pay. The exclusion keeps family members from using the policy to claim against each other.
Care, custody or control: personal liability excludes damage to property rented to, used by, or in the care of an insured, with a standard exception for fire, smoke or explosion. A vacationer who negligently burns a rented cabin can look to personal liability; a water or breakage loss to that cabin is not handled under this coverage.
Damage to property of others is an additional coverage that lets the insurer pay small losses to other people's property caused by an insured, even without legal liability. It helps keep good relations with neighbors and friends. It pays for others' property, not the insured's own, and does not reach damage from an excluded motor vehicle.