The governing body of a county having the county executive form of government that is adjacent to a county having the urban county executive form of government may establish different terms of office for initial and subsequent appointments for those boards, authorities and commissions for which it is given the authority to appoint members, excluding authorities empowered to issue certificates of indebtedness. The different terms of office for such boards, authorities and commissions shall be for fixed terms, and such different terms of office may include, but are not limited to, terms of either two or four years and terms that extend until July 1 of the year following the year in which there is a regular election provided by general law for the election of supervisors. If the board establishes different terms of office pursuant to this section, the new terms shall affect future appointments to such offices and shall not affect the existing terms of any commissioner or member then serving in office. This section shall not affect the removal of any member of a board, authority or commission for incompetency, neglect of duty or misuse of office pursuant to provisions of general law.