Should I still file an insurance claim?

About a couple of days ago, I got into an accident with a neighbor in the apartment complex parking lot. The parking lot is V shaped and can only hold 4 cars. The girl was pulling out of her spot, and was ready to pull out into the streets. I checked my mirrors and saw that there was nothing, so I pulled out (still looking at my rearview as I backed out). All of the sudden, I felt a huge thud, and I couldn't figure out what I hit because there was nothing in my rearview mirrors. I turned all the way to my left, and the girl in the other car was right behind me. My first reaction and question when we got out of our cars was "What were you doing in that area?!?!", she said she was pulling in so she can yield to the oncoming traffic. The next day, I asked her again, she said she didnt say that, she now claims that she was just pulling out of her parking spot. I regret not taking an aerial view of the incident because if you saw it, it will all make sense. Anyways, after talking to her, she told me the estimation to the damages, and I asked if that is exact/final, she said yes. Then later, she said it might be slightly more. When I was trying to get snapshots of the estimation paper (I didn't want to just take that paper from her since its the only copy), I took a copy of the last page which is the fine print, and she was telling me that the last page basically explains what an estimation is. I thought why would she have a problem with me getting a copy of the fine print?!? I called my insurance company to tell them I want to start a claim because this girl is not accurate on what she says. She'll say one thing, and then another. I thought it would be safer going through insurance. They said since damages are only $700, it is better to deal with this privately, and get every single documentation of our actions such as bills, auto repair shops, etc. Overall, I basically felt that the accident could have been prevented. She just moved into the apartment complex, so I dont know her. Would it be better to save money and deal with this privately, or risk increase in my insurance rates and file a claim?