public liability insurance
No, liability insurance is when there are injuries involved. If you are injured in an accident when someone else is driving your car, your liability insurance would cover your medical costs. Comprehensive and collision insurance on the car you were driving should pay for damages to the vehicle.
Yourself and the cargo you are carrying should be covered under this insurance. It also protects your company from liability in case of an accident or emergency.
A public liability is a form of insurance purchased by businesses. This insurance protects a company from lawsuit in the event a consumer is harmed as a result of their actions.
Public liability is a type of insurance policy. This insurance protects or covers the user for legal liability to third parties (that is, potential lawsuits against them imposed by third parties).
Collision protects you and your car if you are in an accident, liability protects whomever you may hit.
Liability coverage (often simply called "liability") protects you from damage you do to others or to property in an accident. Some level of auto liability insurance coverage is required in all 50 states - get info for your state's auto liability insurance coverage requirements here.
Damaging someones property or causing injury or death.
Builders liability insurance protects builders from laws suits. If damage happens while the builder is working on a home it protects him should the homeowner decide to sue. It also protects the builder in the event an employee should injure themselves and decide to sue.
Business liability insurance protects a company's assets from a lawsuit. If a business is high risk or doesn't have enough capital to cover a lawsuit, they should have business liability insurance.
You are subject to liability insurance requirements whether or not an accident occurs. An accident has nothing to do with liability requirements. You are required to have liability insurance of at least the minimum required by your state before getting into a vehicle and driving it. Driving is what triggers the law.
I doubt there is insurance available to you. With a permit, you are actually driving under the authority and liability of the licensed driver in the car. It is their insurance that protects you and other drivers in the event of an accident. Once you have a driver's license and own your own car, you can purchase insurance from almost any insurance company.