Is the money paid to me by the insurance company, after I caused a traffic accident and the fault was shifted to me at the request of the injured party, lawful or unlawful?

Since you are at fault for the accident, you bear full responsibility. If the other party forgives you, that is their right, but it is not permissible for them to deceive the insurance company by making it pay for repair costs, as this is fraud and falsehood. Therefore, you must return the money to the insurance company, and if that is difficult, then give it as charity to the poor and needy. Your financial situation is not a justification for taking money unjustly.