Agreed, you can't file a claim against your own liability insurance. But that is not how liability insurance works.
If you are away from your home and damage the property of someone else or injure someone else, they will sue you. Your homeowners liability coverage will kick in & protect you from the liability of the suit. Example: if you are at a golf course and one of your shots injures someone or takes out a window, they can sue you. Most of the time the insurance will pay without a law suit.
You are not filing a clam against your own policy, it is the injured party making the claim through a law suit.
The AMA is going to make your homeowners liability insurance pony up first, if the damage exceeds the policy limits, then AMA will cover you.
Here is an excerpt form the AMA document
How to file an insurance claim
AMA insurance is “secondary” or “excess” insurance. So in addition to notifying AMA Headquarters of the accident, this also means an AMA member must first file the incident through any other insurance available to him or her: i.e. homeowners, renters, automobile, health insurance, etc. If the primary insurance covers the total claim amount, the member should let the AMA know so that we may close the file.