In addition to the events listed in section 32-560, a vacancy on the board of directors shall exist in the event of (1) removal from any district of a director, (2) removal of a director from a subdivision in which he or she was a director, or (3) elimination or detachment from a district of the part thereof where a director or directors reside. In the event of a vacancy, such vacancy shall be filled by the board of directors. Such appointments shall be in writing and continue for the unexpired term and until a successor is elected and qualified. The written appointment shall be filed with the Secretary of State.