The chef is usually the lead cook in a restaurant. "Főszakács" in Hungarian. A cook can be anyone who prepares food.
So, that's the difference. But maybe it could be accepted.
I agree. In colloquial English, chef often refers to any cook.
(A bit like how in German, to call the waiter's attention, you might say, "Herr Ober!" = Mr Head (Waiter), regardless of whether he's the head waiter or not. And a policeman might be addresssed with "Herr Kommissar", regardless of whether he holds the rank of commissioner or not.)