In addition to any other process for removal provided by law, the incumbent of any appointed office provided for in the Charter or created by ordinance under the authority thereof may be removed at any time after the expiration of three months from appointment by the registered voters of the City. The procedure to effect the removal of the incumbent of an appointive office shall be the same as that for the removal of the incumbent of an elected office by the recall, with the following exceptions:
(a) The petition for the removal of the incumbent of an appointed office shall be signed by registered voters equal in number to at least 20% of the entire vote cast for the office of Mayor at the last preceding general municipal election, or primary nominating election, at which a Mayor was elected.
(b) If a majority of the registered voters voting on the question vote in favor of the removal of the appointed officer, the officer shall be removed from office upon the declaration of the result of the election by the Council, and the office shall thereupon become vacant. The vacancy shall be filled by the appointing power in the same manner as other vacancies, but any appointed officer so removed shall not be eligible to any appointed or elected office under the Charter until the expiration of two years after removal.