All officers appointed by the mayor or council, except officers appointed to fill vacancies in elective offices, shall hold their respective offices until the first Monday of May next after such appointment, and until their successors are qualified and enter upon the duties of their office, unless a different term of office shall be provided in this act, or in the ordinance creating the office. Any officer elected to fill a vacancy shall hold the office during the residue of the term of office in which the vacancy occurred, and any officer appointed to fill a vacancy in any elective office shall hold such office until the next annual city election.