What Levissima has recounted accords with my experience.
In Italy, Wills are necessarily dealt with by notaios, (there is no concept of a 'executor'), and in my experience making your Will a 'public document',(which means making it before a notaio) is not only the most secure way of ensuring your Will is respected unconditionally, but it is probably the most economical.
A holographic Will is entirely acceptable in law, but it still needs to be 'published' by a notaio. If you have already lodged your Will with a notaio (paying them something like €600 for the privilege) they'll knock this off their 'publishing fee' (more or less).
So, if you have a notaio who you trust to be honest and straightforward, just use the notaio.
But, a further consideration; if your English Will starts off with the conventional phrase 'this will and testament revokes all previous...' then watch out: that can be considered to supersede any Italian Will you wrote at an earlier date. So, if you 'update' your English Will, you might well negate your Italian Will.