AMA is not a "privately held company" by any stretch of the imagination.
You want to keep playing back and forth on who is right and who is wrong. The AMA is MOST CERTAINLY a privately held, self supporting, not for profit corporation. If a company is not pubic entity then it's a private entity. The board of directors whose names are held by the NFP can choose to do whatever they want with the company. The board of directors have chosen to set the company up the way it is presently, but it's still privately held.