You raise an interesting area of confusion with nonprofit governance.  With nonprofit corporations, there is often a distinction between officers of the board and officers of the corporation.  Most state nonprofit corporation laws require someone to have the functions of president/chief executive officer, secretary, and treasurer, although they can be called anything the organization wants to call them.  They could be called Grand Poobah, Scribe, and Moneybags if the organization wanted to.

In smaller organizations, these officers are traditionally on the board of directors.  (See Ready Reference Page: “Bylaws Function as ‘Constitution’ of Nonprofit Corporations”)  Often there is a Chair of the Board officer to run board meetings and a separate President or CEO to have day-to-day-full time management responsibility.  Any vice-presidents, the secretary and treasurer are usually volunteer board members but are responsible to oversee the traditional duties of the offices.

In larger nonprofit corporations like hospitals or universities, the president/ceo may or may not be on the board.  The secretary and corporate treasurer are often not members of the board, but still have the corporate responsibilities of these offices to run the corporation.  There is no legal requirement about how it should be done.

If your nonprofit is organized as a trust, and not as a corporation, the members of the governing body are all “trustees” and may not have any of the specific roles required by a corporation.  They might be assigned the specific duties, but again there is no legal requirement about what should be done.

Your choice.