Does the resignation of the President of a nonprofit corporation result in his termination as an incorporator, director or member of the organization when the letter says nothing about these other positions that he holds?
No and probably not, but maybe.
Once a nonprofit corporation has started its activities, the incorporators have no further role in the organization and there is no way to resign from the historical fact that a person served as an incorporator.
A silent resignation such as you describe would not normally cause a termination of membership except in those few situations where the bylaws provide that the officers are the members of the corporation. (This occurs, for example, under the Delaware corporation law where a nonprofit apparently must have members, but the organization doesn’t want members and masks the issue by making the board and the members the same people.)
Nor would such a silent resignation normally cause termination as a director except in those situations in which people are elected (usually by the members) directly to serve as officers of the corporation and by virtue of serving as an officer also have a seat on the board.