What to call a person who runs a non-profit organization?

Solution 1:

The question conflates occupations, sectors, and titles. An expeditor works in a kitchen but is neither a chef nor a cook. An office manager who works for a restaurant is as much an office manager as one who works for a law firm. And the person who designed their website is on some level a web designer, but might be an "intern" or "Senior Director, Digital Marketing" on the résumé.

It is relatively rare that a title implies an industry, except in specialized roles in realms like the government, military, or religion—if you are an archbishop, most people won't ask whether you are archbishopping for an accounting firm or for the electrical utility.

A chief executive is a chief executive. It doesn't matter whether she works for the local food bank, the municipality, or a regional chain of veterinary clinics, and it doesn't matter whether her title is General Secretary or County Manager or President and CEO. If you want to say she is director of a non-profit, you have to say she is director of a non-profit. There are no shortcuts.