Translation:The famous Italian cooks are cooking in these new hotels.
If a cook is famous, he is called a chef.' not a cook. A cook works in a greasy spoon restaurant.
You are right that in English a cook is not a chef. A chef is a head cook who supervises other cooks or has distinguished himself/herself in some way
Hungarian has its own word for chef, namely 'séf'. The exercise sentence here uses the word for 'cook', not the word for 'chef'.
However, I think you go too far when you insist that there cannot be such a thing as a 'famous cook' or 'famous cooks'. For example, suppose a famous French chef brings his entire kitchen staff to an exhibition in Budapest for a one-day event. It is reasonable to call the chef and his subordinates 'the famous French cooks'.
Every chef is a cook, but not every cook is a chef.
No - a cook can also work under a chef. A "chef" is a specific job - not just a famous cook. Generally avoid it in Enlish unless they really are a "chef".
how about :the famous italian cook cooks in these new hotels...? The choices of answers in language learning in beta are very few