Property coverage pays for damage to the insured's own property. Liability (casualty) coverage pays others when the insured is legally responsible for harm to them or their property. If the insured negligently knocks down a neighbor's fence, that is a liability loss, not a property loss.
Who is claiming? Under California's claims regulations:
First party claimant (10 CCR §2695.2(f)): asserts a right under the policy as a named insured, other insured or beneficiary.
Third party claimant (10 CCR §2695.2(x)): asserts a claim against a person or interests insured under the policy.
Subrogation: after paying a loss, the insurer takes over the insured's right to recover from the party at fault. The California standard fire form lets the insurer require an assignment of that right only to the extent of its payment, so it cannot recover the part it never paid, such as the insured's deductible.
Settling disagreements:
Arbitration: a neutral third party hears both sides and decides the dispute outside of court. (A mediator only helps the parties reach their own agreement.)
Appraisal, under the standard fire form: narrower. It applies when the insured and insurer fail to agree on the actual cash value or the amount of loss, not on whether a loss is covered.