That the term of office of officers and commissioners appointed by the governor, in cases not otherwise provided, or where no term is specified in the act creating such office or commission, shall expire 2 years from the first day of January of the year when the appointment is made, unless the appointment shall be by the commission limited to a shorter term, in which case it shall cease as limited, or unless the appointment be to fill a vacancy, in which case it shall continue for the remainder of the term: Provided, That in cases where by law the office does not expire with the term, such officers shall hold the office and continue to act until their successors are appointed and have qualified.