I tend to make claims only when required as in to the tune of several thousand. I have always paid for any small damages as these do add up in your insurance history. Even if you are not at fault, you become high risk if too many claims are made and yes the rates will go up. Cannot comment on your incident however if it was a scratch on the bumper I would simply settle for $500 or so have have the paint corrected, if the damage is more severe, then do get insurance involved.
Bought my daughter a used Yaris when she turned 16, told her to be careful as all accidents that were her fault are not covered and I will need to repair the car at my expense. I did take out liability insurance which was only a couple of hundred compared to the $2000 a year to cover her car due the the young age group. We do have no fault insurance here. Car was only worth $5000 so I self insured at least for damages to her car, a risk I was willing to take and could afford. She drove the car for 5 years with no accidents except for curb rashed wheels. Saved $9,000 in insurance.
Same goes for my classic Mercedes SL which is garage kept during the summer, taken off the road for winter storage from November until May. Only driven if the weather is nice therefor hardly ever used. A 1996 with a whooping 38,000 miles on the clock. Liabilty Insurance and plates cost around $300 a year. I have had the car for 18 years now, insurance wanted $1200 a year. Due to the extremely low risk factor, I once again self insured.
Just so people understand how our no fault insurance works. If I am involved in an accident and NOT at fault the other persons insurance covers his car and mine therefor if I was in an accident with the Mercedes and it was the other persons fault, everything is taken care of by the other persons insurance. If I were at fault, the liability part takes care of the other persons damages however they will not fix my car since I declined full coverage.