Ah the "Chef' business....
Almost as much ink has been spilled on this subject as "what is the "Best" knife?"
I come from a planet far, far away... There were no cooking shows on TV that glamorized cooking back then, and cooking wasn't really all that interesting anyway.
Funny thing was, back then, at every place I worked at, the only person you called "Chef" was the guy who hired you, the guy who handed you your paycheck, and the guy who gave you orders. Everyone else were "cooks" and there was no shame in calling yourself a cook.
Then I went to Europe and worked under some really big shots, highly decorated Chefs, in charge of Brigades of over 40 cooks, multiple F & B outlets. But we never addressed the Chef as "Chef". If and when we talked to him, it was "Herr" (Mr.) so-and-so. In France it was"Msr." so-and-so. Even the owners of the hotel would address the Chef as "Herr". And even back then, the custom in Europe was to list your profession next to your name and address in the phone book, and every big shot Chef--the guys in charge of brigades of over 40 cooks, the 5 star hotels glamorized by Hollywood, would list their profession every year in the phone book as "Koch" or "cook", even though these guys were well over 50 and earning some mighty big bucks.
So for me, the whole "Chef" thing isn't about the title, or what your co-workers or employers call you, it is how you are judged:
-A cook is judged by what they put on a plate. If it's good, they stay and put out good plates. If it's bad, they go.
A Chef is judged by how well s/he runs the kitchen. If the kitchen makes money, the Chef usually* stays. If the Chef can't keep the kitchen above water, they never stay for longer than a month and they go. Sometimes less than a week.
Harsh judgement, but very realistic. We are in the business to feed people. If we can't make money, the employer turfs us, and you really can't blame the employer for doing so. It's a business, not a charity.
And that is the true meaning of a "Chef", a "Chief" in any other language. The person who is tasked with making a profit for his employer, with food, labour, and overhead costs provided by the employer. This model fits in with a sandwich shop or a taco stand as much as it does with a fine dining establishment.
I am proud to call myself a cook. I did a classical apprenticeship as a cook, sweated my privates off for three years working double split shifts and going 1 day per week in trade school and getting paid half of what the dishwashers were getting. Hard to buy uniforms, books, and knives with that salary but I did it. When I finished and got my "papers", my profession was listed on the documents as "Cook" in all three of Switzerland's official languages, so there was no doubt in anyone's mind that I was and still am a Cook.
*Just because the Chef can make money doesn't guarantee that s/he will stay. The employer can get greedy and turf the Chef out, give his Sous a new embroidered jacket and a stack of business cards that say "Chef", pay him a buck an hour more, and when that guy fails or asks for more money, turf HIM out and give the next guy an embroidered jacket.........