Resignation from the purchase contract as a private seller

Hello, I signed a purchase contract for the sale of my car 5 days ago. I did not accept a deposit and the car is not registered on me but on my mother, I told the buyer about the sales authorization I received from her, but did not give it to him. Yesterday I bought a new car and, since I thought I could withdraw from the 1st contract, signed a new purchase contract in which I sold my old car. I I got in touch with the dealer of the 1st purchase contract and wanted to ask him to withdraw from the contract, but he said that he was not ready to do so. So I talked to the dealer of the 2nd purchase contract and explained my situation to him, who in turn said that, since I am a private seller and did not accept a down payment, that car also does not have to sell to the 1st buyer. Is that true? And if so, any idea, as I might get out of there after all Come?