The acceptance by a nonresident motor carrier of the rights and privileges conferred by the laws now or hereafter in force in this State, permitting the operation of motor vehicles as evidenced by the operation of a motor vehicle by such nonresident either personally or through an agent or employee on the public highways in this State, or the operation of such nonresident either personally or through an agent, lessee, or employee, of a motor vehicle on the public highways of this State other than as so permitted or regulated, shall be deemed equivalent to the appointment by such nonresident motor carrier of the Director of the Department of Motor Vehicles, or his successor in office, to be his true and lawful attorney and the attorney of his executor or administrator, upon whom may be served all summonses or other lawful process or notice in any action, assessment proceeding, or other proceeding against him or his executor or administrator, arising out of or by reason of any provisions in Chapter 31 of Title 12 relating to such vehicle or relating to the liability for tax with respect to operation of such vehicle on the highways of this State. Acceptance or operation shall be a signification by such nonresident motor carrier of his agreement that any such process against or notice to him or his executor or administrator shall be of the same legal force and validity as if served on him personally or on his executor or administrator. All of the provisions of Sections 15-9-370, 15-9-380, and 15-9-350 shall be applicable with respect to the service of process or notice pursuant to this section.