To me, it is a person who has been in the cullinary field, back of house, for a number of years to have a well rounded knowledge on how to run a multi-stationed brigade style kitchen, do the human relations side of running a crew of (for lack of a better explanation), Line, Sous, Garde Manger, Prep, Washers and sanitation, cost your food, order the supplies, plan events, do purchasing, know how and when to upgrade facilities, present a positive image of your establishment to the customers, front of house, and owners, etc... Coordinate with FoH, Run the FoH staff through the menu, give pointers... Did I mention training? Biggest part is being the instructor for your BoH (and some FoH) staff, so they know what the standards are and hold them. Rigidly. She (or he) also has to enforce discipline, and usually makes the ultimate hiring / fring decisions for BoH (and again, sometimes FoH). Sometimes, the Exec will run the entire foods operation of 3 or 4 rooms/kitchens in a facility. When they HAVE that knowledge, skill, and ability, it's not a bragging rights, you can sense it in someone.
Food school, while giving you a good background knowledge, does NOT a "Chef" make, no more than having a fresh out of school Bachelors in Business will make someone a Fortune 500 CEO or hedge fund manager. I've been in and out of foods for decades, and while I can do an awful lot of what does a Chef (and (might) pull it off in a small joint) and learned a lot of thsoe skills in and out of kitchens, I wouldn't call myself a "Chef".
Of course, this is all my opinion, but having worked for world class Exec Chefs, some aspiring to Michelin star-dom, you can sense if they have it (or are it), ore are posers.