Ok, I think I need to stop the misconceptions here before they get completely out of control. In my former life I was a casualty adjuster for motor vehicle accidents.
1) Yes, you should always contact the police when you are involved in an accident, as the police report usually prevents fraudulent claims from happening. That said, unless it is an accident with injuries the police don't always come (especially in big cities), and a police officer is not required to file a report unless there is visible damage over a certan dollar amount (depending on state), or injuries;
2) The amount you pay for premiums is contingent on many things, your driving history certainly one of them. If you are involved in an accident, the amount paid out for property damage will cause your premiums to increase if you are at fault for the accident -- and the dollar amount certainly does matter. A $600 claim is not a huge increase, a $10,000 claim will slam you. The citation you received from the police is going to hurt you also in the premium department as it affects your driving history. Note that the increases do not go into affect until your policy renews, and you can't hide from it -- every insurance company has access to the registry of motor vehicles, and every single claim you make is reported to a claims clearinghouse database. Insurance carriers are all knowing in the claims department, so always disclose and never lie about it as it is grounds for policy cancellation;
3) Always alert your insurance carrier to fender benders, even if claims are not filed. Your insurance carrier cannot defend you if you do not make them aware of the accident. In most states the statute of limitations on personal injury claims is 2-3 years, and a year out is usually the time when we receive a notice of appearance from claimant's counsel advising of the personal injury claim. You never tell us about the accident, then you removed our right to investigate the accident, interview the occupants of both vehicles, seek out witnesses in the area, and secure photos of both vehicles TO PROTECT YOU. Our job is to adjust claims and protect you from exposure to excess judgements (which is amounts awarded by juries for sums that far exceed what your coverage is). The insurance carrier's exposure is what you purchased for coverage. You on the other hand have exposure for everything you own and attachments on future wages. You purchased insurance for a reason, so use it. Remember your premiums will not go up unless a claim is filed and you are at fault for the accident. If no claim ever gets filed, no worries on your end. Further, not advising your insurance carrier of accidents promptly is grounds for cancellation or non-renewal of your policy.
4) I am sorry, but I have to laugh at loud at an assertion that adjusters count on ignorance, or take marching orders from our insureds. Not the case I can assure you. We are extremely well trained and educated in the field of insurance, and adjust claims in good faith and based on very strict regulations. Insurance carriers are audited yearly by the state -- trust me, they aren't screwing people to save money. They will however complete an settlement evaluation with a high and low figure range, with low pretty much being reflective of the information they have on hand. High taking into consideration some what if scenarios. If you don't like the figure presented then you are going to need to provide more commentary other than you don't agree with it. You will need to be prepared to submit solid reasons as to why you feel it should be higher. Note that 90% of my day was negotiating with attorneys who are extremely good at the art of negotiations. An insured would not intimidate me in the slightest. Give me reasons and solid arguments why you should get more and if I agree with you I will give you more. It really is that simple;
5) An appraiser is the party who appraises the damage to your vehicle or damaged property. While they could be employed by the insurance carrier (most are private contractors), they are licensed by the state and are bound by very strict regulations. Unless the vehicle is at a licensed autobody shop, and broken down in front of them, then they are only allowed to write for damage they can see. Once the autobody shop starts breaking down the vehicle they call them back to write for the hidden damage. For a vehicle to be considered a total loss it has to reach a certain threshold of dollar damage (which I think is 75% of the value of the vehicle), or it has to have been submerged in water, or the frame completely bent and unrepairable. They are not going to risk losing their license to play fast and loose for an insurance carrier. An adjuster is the party who works for the insurance carrier that determines the dollar value of your claim (property damage and injury -- two separate claims).
As to the opening post -- I am very sorry you had this accident today and am very glad to hear you were not injured.