When you express the occupation with fare you're stating a role, and that's why the definite article: it's close to "my cousin plays the lawyer", meaning he has the role of lawyer (in that case, in a play). When you're using essere instead, just like in English, you use an indefinite article because there are many lawyers.
Apologies if I have missed this in your comment, but you don't seem to have answered the question: why can't it be "THE lawyer" (when you're talking about a specific lawyer)? Would this also use essere?
Btw, congrats on the 1609-day streak, I'm not even sure how that's possible :o
Above sentence was the only correct solution to 'My cousin is a lawyer'. My apparently incorrect answer was 'Il mio cugino è un avvocato'. I saw f.formica's post and thought I now understood why my answer was incorrect.
But then I had to translate and put the words in the correct order for the sentence 'My uncle is a presenter'....
[zio] [ingegnere] [operai] [un] [è] [Mio] [conduttore] [postini] [contadino]
Now why is my answer 'Mio zio è un conduttore' this time correct and did I not have the possibility to answer 'Mio zio fa il conduttore'?
Can you use for certain professions only the verb 'fare'?