Making a claim with “the other driver’s” insurance company? That’s a third-party claim. Driving a company car and sustain an injury? You’re making a third-party claim there, too. Injured while driving your own car in the course of your job? A third-party claim with your employer’s insurance company would be your mechanism for recovery of medical and repair costs. Any claim made to an insurance company other than your own is considered a third-party claim.
If you have paid off your car, comprehensive coverage is optional. It may be a good idea to find out the Kelley Blue Book value of your vehicle. Would you be able to pay that amount to repair or replace your vehicle if it were stolen or damaged in an accident? If you can't afford to pay much out of pocket, then buying optional coverages, like comprehensive coverage and collision coverage, may be a smart investment.