(a) A vacancy on the commission shall be filled for the remainder of the unexpired term as follows:
(1) if the vacancy involves a member appointed by the county judge or a county commissioner, the vacancy is filled, except as provided by Subsection (b), by appointment of that officer or the officer's successor in office; or
(2) if the vacancy involves a member appointed under Section 161.055(a)(2), the vacancy is filled as provided by that section for an appointment to a full term.
(b) If the county judge or county commissioner, as applicable, does not fill the vacancy before the 60th day after the date the position becomes vacant, the commission may fill the vacancy by a majority vote of the remaining members.