Any county board when providing for the reapportionment of its county under this Division may provide that the chairman of the county board shall be elected by the voters of the county rather than by the members of the board. In that event, provision shall be made for the election throughout the county of the chairman of the county board, but in counties over 3,000,000 population no person may be elected to serve as such chairman who has not been elected as a county board member to serve during the same period as the term of office as chairman of the county board to which he seeks election. In counties over 300,000 population and under 3,000,000 population, the chairman shall be elected as chairman without having been first elected to the county board. Such chairman shall not vote on any question except to break a tie vote. In all other counties the chairman may either be elected as a county board member or elected as the chairman without having been first elected to the board. Except in counties where the chairman of the county board is elected by the voters of the county and is not required to be a county board member, whether the chairman of the county board is elected by the voters of the county or by the members of the board, he shall be elected to a 2 year term. In counties where the chairman of the county board is elected by the voters of the county and is not required to be a county board member, the chairman shall be elected to a 4 year term. In all cases: (i) the term of the chairman of the county board shall commence on the first Monday of the month following the month in which members of the county board are elected, and (ii) no person may simultaneously serve as a member of a county board and the chairman of the same board if the office of chairman is elected by the voters of the county rather than by the members of the board.