Dioxide45, thanks very much for the information!
I've looked into this topic some during the past day, and have found some additional information.
According to a poster on another forum, credit card companies usually don't tell your regular auto insurance about the claim, even if the credit card coverage is secondary. Instead, the credit card company just asks to see the declarations page from your auto insurance to determine what coverage you have. If you have collision insurance, the credit card company pays just the deductible and it is then up to the renter to decide whether to file with their regular insurance to cover the rest of the damage, or pay for the rest of the damage out of pocket without filing a claim with their regular insurance. If you have NO collision insurance, the credit card company pays for all the damage to the rental car, up to the amount of their coverage limit. (If you totaled a $80,000 sports car, say, that would likely exceed the maximum benefit on the free coverage.)
The above policy (on notifying your regular auto insurance about the claim) may vary between different U.S. states. However, I'm not sure which state's policies are the determining factor -- the state where the car damage occurred, or the state where you have your auto insurance.
Also, when a renter damages a car, they may owe charges in addition to the actual repair cost of the car. There is often a fee charged by the rental car company for administering the repair, plus a fee charged by the rental car company for "loss of use" of the car while it is in the shop. Personal auto insurance may not cover these fees. Coverage through a credit card may or may not cover them, depending on the particular card's policies.
When we had our recent claim with Amex for damage to an Alamo rental car, Alamo charged an administration fee, which Amex paid. Alamo also had a "loss of use" fee, but Amex negotiated with Alamo and Alamo removed that charge. We paid nothing at all for the damage.
It's also occurred to me that using RCI Points for a car rental might not provide any automatic car rental damage coverage. (There is a $25 transaction fee when booking a car through RCI Points, but I don't know if that fee, when charged to a credit card, qualifies for free rental car damage insurance.) So, even if I have extra RCI Points, I won't use them for car rentals.