Thank you to everyone who weighed in with their experiences. It was very helpful!
The damage repair estimate came in at just under the value, so the car was not totaled. So we get a check for that, less our deductible, on Monday. Unfortunately, I did file a claim under my insurance first because I thought I was supposed to. There is also a claim against the driver's policy. I hope my premiums don't go up.
Fortunately, the other driver has the same insurance company, so we should get our deductible back more quickly, as long as liability isn't an issue. The police responded, but I don't know what kind of report was taken, we have to make a request for that. The damage and position of the vehicles make it abundantly clear that he backed out into our moving vehicle (damage starts at center of front passenger door and continues into back passenger door).