More on the insurance aspect- educate yourself ahead of time about what coverage you have. In addition to personal coverage (if you own a car), your employer may have coverage that protects you when renting a car on company business. (If you have an old car with no physical damage coverage, you're still probably covered under the Liability section for damages to cars you rent.) That's in addition to whatever credit card coverage applies, although the latter is usually excess over your personal coverage. (In other words, they hit your regular insurance company up for the money first then go to the credit card issuer if that's not enough). This means that 99% of the time, your regular insurer WILL hear about the claim.

I'm an actuary in the property-casualty side of the insurance business so I have fun playing with the drones behind the desk who try to tell me why I have to buy their stupid "collision damage waiver" (which they aren't allowed to call insurance). They interrogate me about my personal coverege and usually shut up when I tell them we've got a million-dollar umbrella liability cover on top of regular limits.