Fault is either admitted or proved. Determining fault may involve a lawsuit. The insurance company of the at-fault driver pays the losses of the other driver. Collision covers property damage to your car caused by an accident. It does not cover the other person's car.
Liability insurance covers property damage and personal injury to the other driver.
Medical insurance covers the cost of your personal injury. Your health insurance company also may cover the cost of your medical care resulting from an automobile accident. If you are in a no-fault state, it doesn't matter so long as your insurance policy is adequate to pay for all losses. However, if your losses exceed the value of your policy, then you have a problem. In an at-fault state, it creates a problem for the innocent driver.