California's Fair Claims Settlement Practices Regulations define the two kinds of claimant:
First party claimant (10 CCR §2695.2(f)): a person asserting a right under a policy as a named insured, other insured or beneficiary. It expressly includes a person seeking uninsured motorist benefits, even though another driver caused the injury.
Third party claimant (§2695.2(x)): a person asserting a claim against an insured or the insured's interests.
Subrogation: the party who paid a loss steps into the injured person's rights against the party at fault. Example: a delivery driver hit by a negligent motorist collects workers' compensation. Under Labor Code §3852(a), his compensation claim does not end his right to sue the motorist, and the employer who paid, which includes the insurer (§3850(b)), may also claim against the motorist. An insurer pursues the party at fault, not its own insured.
Arbitration: a neutral party decides a dispute outside of court. Ins. Code §11580.2(f) requires the auto policy to provide that disputes over entitlement to and the amount of uninsured motorist benefits are settled by agreement or, if the parties disagree, by arbitration before a single neutral arbitrator.