How does the lack of articles alter sentence meaning in Italian?

I am trying to learn basic Italian. One thing that, as a native English speaker, I find very confusing is the apparent lack of a need to supply in/definite articles.

So for example take the phrase "Io cucino il pollo". The literal translation of this is "I cook the chicken". However, I'm given to understand that the actual translation is "I cook chicken". Another example might be "Mangio formaggio" which literally translates as "Eat cheese" but which is understood as "I eat cheese".

While these differences may seem small in English the sentence structure does render them subtly different. "I cook the chicken" is a statement which implies a definite chicken to be cooked whereas "I cook chicken" is more of a question or a generalisation.

Do these subtle differences just not exist in Italian? Or is there another way of phrasing the sentences to make the difference understood? Either way, is there any consistency or rule here I can use as an aid to learning and remembering?

There is another issue in your examples that complicates the problem, and it is the difference between present simple and present continuous. I believe that there are three different concepts that you may want to express; here they are with their most idiomatic translations (in my opinion).

• I cook chicken, in general, as a habit: Cucino il pollo. As you noted, in Italian definite articles are often used in a generic meaning that is not present in English.
• I am cooking chicken, in this specific moment (but it doesn't matter which chicken): Sto cucinando il pollo, or also Sto cucinando un pollo, or also less formally Cucino il/un pollo. In Italian the present simple is often used also in this sense, especially in spoken form. Sto cucinando pollo, without an article, is less common but it sounds acceptable to me, especially in the form of an answer: Cosa stai cucinando/Cosa cucini? Pollo.
• I am (now) cooking the chicken; I mean, that specific chicken that we bought yesterday: Sto cucinando quel pollo, or Sto cucinando il/quel pollo che abbiamo comprato ieri.