Even in english you would not say "I am the cooker." or "He/she is the cooker." You would or could however use the word "cook" as a noun, referring to someone who cooks, as I assume is the case with the word "cuoco". I am more curious as to whether the word cuoco is the same as the word for chef. In english a true chef would be very offended if someone were to refer to him/her as a "cook".
"I am a cook" is the possible answer in the question "What is your profession?" - i mean that it imports a capacity: I am a dancer, I am a teacher e.t.c. On the contrary, "I am the cook" is a much more rarely used sentence,ex "I am the cook in Lio's resturant" "I am the cook here, so do what i tell you!" because of the definite article. Is this exactly the same in Italian? Io sono il cuoco cannot be the answer in the question "what is your profession"? "Io sono un cuoco" is the right answer?