The requirements as to security, proof, and suspension in sections 60-507 and 60-511 shall not apply:

(1) To the operator or the owner of a motor vehicle involved in an accident wherein no injury or damage was caused to the person or property of anyone other than such operator or owner;

(2) To the operator or the owner of a motor vehicle legally parked at the time of the accident;

(3) To the owner of a motor vehicle when such motor vehicle was being operated without the owner's permission or consent at the time of the accident; or

(4) If, prior to the date that the department would otherwise suspend license and registration or nonresident's operating privilege under sections 60-507 and 60-511, there shall be filed with the department evidence satisfactory to it that the person, who would otherwise have to file security and proof, has (a) been released from liability, (b) been finally adjudicated not to be liable, (c) executed a warrant for confession of judgment, payable when and in such installments as the parties have agreed to, or (d) executed a duly acknowledged written agreement providing for the payment of an agreed amount in installments, with respect to all claims for injuries or damages resulting from the accident.