A vacancy in the council shall be deemed to exist in case of the failure of any person elected thereto to qualify on or before the date of the second regular meeting of the new council, or by reason of the death, resignation, removal from office, removal from the city, continuous absence from the city for more than three months, or conviction of a felony of any such person whether before or after his/her qualification, or by reason of the failure of an council member without good cause to perform any of the duties of membership in the council for a period of three months. In each such case the council shall by resolution declare such vacancy to exist and shall forthwith appoint an eligible person to fill the same until the next regular municipal election, when the office shall be filled for the unexpired term; provided that any vacancy resulting from a recall election or from a resignation following the filing of a recall petition and any vacancy in the office of mayor shall be filled in the manner provided by this charter.