You need to report every accident, no matter how slight, to your own insurance company.
If your credit card provides primary rental car damage coverage (few do) you mention that to your own insurance company which, in turn, should not pay a claim immediately. The theoretical course of action is for the other party to come after you first but in today's world, it is more convenient to file the claim with your insurance company first. In this instance, when your insurance company denies the initial claim, the other party then comes after you and you refer it to your credit card company.