Does a rental car have liability insurance

Does personal liability insurance cover damage to a rental car?

While moving, I accidentally touched the rear-view mirror of a VW T4 with my rental car (Mercedes Vito bus), which then jumped out (i.e. not completely driven off). There was no damage to my vehicle.

The driver of the T4 was right there when it happened and then wanted to pay me 500 € for the "damage", which actually wasn't damage. But there was obviously also another damage to the mirror, which must have happened before through own fault (front, deep plastic scratch, opposite direction).

Since we couldn't come to an agreement, we called the police. This then mediated and regulated the exchange of contacts and pointed out that this is actually no real damage, but the mirror can be hooked again - we should clarify everything else between us.

My rental vehicle was then taken back from the rental without any complaints on the same evening, I had not reported the accident because the woman wanted to contact me directly and no obvious damage had occurred.

But now, four weeks later, the rental company contacts me and says that a woman came into the shop with a bill and now wants money from the rental for the mirror. I'm supposed to take a stand and pay the bill.

So far so good, but I suspect that the nice woman wants to blame me for another damage to the mirror. How do I proceed now? Report to my insurance company? Does liability take over the settlement of claims? The woman addressed the rental company directly and not directly to me - and they didn't know anything because obviously no damage had occurred.

Thanks for the answers ...