"The insurance company will evaluate the price with their estimator, and issue you a check for the amount off the computer estimate, minus your deductible, in your name"...
this is true if you go through your own insurance but you would be foolish to do so. Submit the claim through the insurance carrier of the party that hit you. Get a couple of damage estimates from independent repair facilities so that you can compare it to the estimate provided by the other parties insurance carrier. If in the same ball park, no problem. If they low ball you show them the other estimates and ask why thier's is so low. Once an estimate has been "agreed" upon, they should send the check to you, the OWNER of the vehicle. They have the right however to make the check out to you and a lien holder (finance company) if one exists and they so desire. Now if you do go through your own carrier and they issue the check to you and you cash it and do whatever you want with it, it's not a problem unless you once again sustain damage to the same part of the car and you try to file a claim for it. Unless you provided them with proof of repair from a facility of YOUR choice, (or take the car to them so they can verify and take pictures of repaired area) they will subtract from your new claim the value of the previous claim. I should note, this is what applies in New York State and yes, sometimes there are differences that occur for various reasons, ie; a mistake - be it honest or dishonest, a less than knowledgeable insurance adjustor, a stupid insurance adjustor, an insurance adjustor that was bribed in one way shape manner of form.....lol. Oh and for the record, I was a Senior Claims Examiner for Liberty Mutual Insurance Company for 5 yrs and "I PLEAD THE 5th"!!!!!