The Council shall at the time of its organization by majority vote, elect from among its members, one as President of Council and another as Vice President of Council, who shall preside at meetings of Council in the absence of the President of Council. The President and Vice President of Council, neither by virtue of holding such offices nor in performing the duties of Acting Mayor, as provided herein, shall be deprived of their powers and rights or be relieved of any of their duties or obligations as members of Council.