The Mayor may remove any member of a board or commission, except a member appointed by Council. The member recommended to be removed shall be entitled to:
(1) An administrative hearing before the Mayor with a statement of the Mayor's reasons for removal being presented. Such member shall have an opportunity to respond.
(2) An appeal to Council, if an appeal is filed in writing with the Clerk of Council within five (5) days of the receipt of the Mayor's order of removal. In such event, Council shall conduct a hearing within thirty (30) days and vote whether or not to affirm the order of removal by the Mayor. A vote of two-thirds of Council is required to overrule the order of removal by the Mayor.