Upvote:3
Not sure whether this fully addresses your question but the “stamp” mentioned by the consulate is that of the local municipality. The consulate will want to see it, if you want this “letter” (attestation d'accueil) to count for something in support of your application.
But if you do get a visa on your own, without invitation, there is nothing illegal about staying with some friends in France. You can make reasonable changes to your itinerary and foreigners are not required to stay at hotels or, in fact, to register their address in any way, whether for short-term or long-term stays.
The text from the French consulate in Houston also implies the invitation could be examined at the French border but that's only because they assume you would enter the Schengen area directly. It should really read “at the external border of the Schengen area”. Theoretically, Italian border guards could thus ask to see it too, if you tell them the main goal of your trip is a visit to friends or family in France.