I heard that repossessing a stolen property without going through police can be considered theft on its own, so how can the owner of a car that was stolen and reported to police as stolen can legally take possession of the car if it is found by the owner abandoned or parked somewhere else? Besides reporting to police that the car was recovered (thus avoiding getting pulled over on a felony stop and get arrested for it), is it absolutely necessary to have the car recovered through the police?
Your evidence of registration of the car would suffice to show that you are not receiving stolen property, although you are correct that not informing the police that it has been recovered could get you pulled over if the car has been reported as stolen.
Given the circumstances of the question, I don't address the means by which the car is repossessed. Not every use of force would be justified to secure the return of the stolen car.
Reporting the theft and recover to the police would also make it easier for you to make an insurance claim.
Repossession of a car stolen by fraud (tricking you into signing over the car title to them in exchange for Monopoly money in an envelope that you don't check until after they are gone, or for a forged check), which is then sold to a bona fide third party individual for value, however, might constitute car theft.