Since you paid out of pocket, there's no point in reporting the car accident yourself. It gets a little trickier when you consider the other party: they could still file a claim against you and you would get stuck with an accident on your records.
Fortunately, most insurance companies use only use "soft inquiries" and do not look at a full credit report. There is a statistical correlation between your credit score and how likely you are to file a claim.
While your old insurance card will show your most recent policy term, a letter of experience will provide your new insurance company with additional details from your old policy. This is usually because your new company is providing a hefty discount for the fact that you were maintaining insurance coverage prior to the start of your new policy.