Vacancies in Council shall be filled for the unexpired term by an appointment made by a majority vote of all remaining members of Council. Such appointment shall be made within thirty (30) days after the next regular meeting following receipt of official notification of such vacancy. If Council fails to fill the vacancy within said thirty (30) day period the Mayor shall make the appointment. Any Councilman appointed under the provisions of this section shall have all the rights, duties, and obligations of any elected Councilman. Wherever this Charter refers to Councilman or elected Councilman, it shall be deemed to include both elected and appointed Councilmen.