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.
Comprehensive car insurance covers damages from an "act of God," or events that are not caused by a car driving into something else. An "act of God" can include things like damage from a heavy tree branch falling on your car. Since you have no control over when or why a tree branch would fall on your car, this kind of accident would be covered under your comprehensive policy.