(a)(1) Any person appointed to office in any city or town may, for cause, after a hearing, be removed by the officer making the appointment.
(2) The council of the municipality may remove, by a two-thirds vote of all those elected to the council, any person in the several departments for incompetency, malfeasance, misfeasance, or nonfeasance in office and for conduct detrimental to good order or discipline, including habitual neglect of duty.
(b) Notwithstanding subsection (a), in municipalities having a population of less than 12,000 inhabitants, according to the last or any subsequent federal census, the mayor may vote on the removal of any person appointed to office in the municipality pursuant to subsection (a) and the mayor shall be considered as a member of the council in determining whether there is a two-thirds vote of the council for the removal of the officer.