Perhaps a board member could be elected president or vice president and then will no longer be president or vice president when that person is no longer a board member.
Anyone who is eligible according to the bylaws could be elected. In this case, the question remains on whether an eligibility requirement is to be on the board.
In RONR, it's one member, one vote. Since your rules allow for something different, it will be up to your association to determine the details for that situation.
You could give notice of this proposed rule and at the meeting, adopt it by a two-thirds vote. Or alternatively, this rule could be adopted by a vote of a majority of the entire membership.
In RONR, the minimum officers for conducting business are a presiding officer and a secretary. Other required officers would be specified in your bylaws.