Hi everyone. I'm glad to say I'm able to close this thread with good news. Originally the other driver lied and said I was the one who hit him, but I just got a call from the insurance adjuster saying he confessed to lying and admitted he hit me. I imagine this doesn't happen very often, so I thought I had better record it here - at least there was one instance! Thanks again to everyone who answered my original questions and contributed to this thread.
I have had this happen to me a couple of times. When the other party has admitted their fault it certainly makes me feel more charitable towards them.
In one case, I was rear-ended while waiting at a stoplight, and the other party admitted to the police that he had been distracted and hadn't noticed the red light, etc. I felt really sorry for him because his wife was really nasty on the phone when he called her. (No, wasn't trying to listen - she was really LOUD.)
In another case, the other party admitted to the police that he had pulled away from the curb without looking out of his side window; he'd glanced in the rear view mirror, not out the window. He was issued a citation.
In both cases I could say kind words to the people before leaving the scene, as they were very polite and honest about what happened.
(I've also experienced cases where the other party was... less agreeable. It's annoying when people hit your car and then wonder why you're making a fuss.)
I am glad that the other party in your case eventually told the truth. These things are stressful under the best circumstances. I hope things go more smoothly for you from here.