Did I become joint owner of the motor vehicle at the time the letter of joint ownership was notarized?

You cannot become joint owner of the vehicle unless your name appears on the title to it. The hand written letter which you feel gives you ownership when the owner dies, may only do so if it meets the requirements of a Last Will and Testament under state law. If by chance it does, but the car is subject to a lien from a lender, you would own it subject to the loan being paid off. You should review this document with an attorney.