(a) All elective officers, including ward officers and school commissioners, shall hold office from the 15th day of March of the year of their first election, for the term for which they are elected, and until their successors have been elected and have qualified.

(b) All appointive officers and all other officers whose terms of office are not herein or by general law otherwise specified, shall, except as herein otherwise provided, hold their offices for the specified terms of their appointment from the first day of April of the year of their appointment, and until their successors are appointed and have qualified.