Except as otherwise provided by State or City law, the terms of members on boards and commissions shall begin on July 1 and end on June 30 of any particular year and the Mayor is the appointing authority for members of boards and commissions, subject to the approval of the City Council. However, if the Mayor fails to make an appointment within 45 days after a vacancy is created or if the Council twice fails to approve the Mayor’s appointment of the same individual to a specific board or commission, the Council shall appoint an individual to fill the vacancy for the remainder of the applicable term. Notwithstanding this provision, the Mayor remains the appointing authority for the purposes set forth in Section 5.09 of this chapter.