Biennially at the first meeting of a new council, or periodically, the members thereof, by majority vote, may designate one of their number as mayor pro tempore or deputy mayor for such period as the council may specify, to serve in the absence or temporary disability of the mayor; or, in lieu thereof, the council may, as the need may arise, appoint any qualified person to serve as mayor pro tempore in the absence or temporary disability of the mayor. In the event of the extended excused absence or disability of a councilmember, the remaining members by majority vote may appoint a councilmember pro tempore to serve during the absence or disability.