You will need an IDP from AAA to rent in Italy.
You will never need an IDL from IDA as it is not a recognized document. It's a scam as per the above post.
https://mominitaly.com/international-driving-permit-italy/
Some European countries require US drivers license holders to have IDP with them. (Austria, Bosnia-Herzegovina, Croatia, the Czech Republic, Germany, Greece, Hungary, Italy, Poland, Romania, Slovenia, Slovakia, and Spain) - however, enforcement by the car rental agencies various depending on country, although it's the police that will fine you - if you get stopped and are asked for it.
Note: Italy in particular allows a sworn translation into Italian in lieu of an IDP.
As far as the unauthorized companies that provide the IDP, although they are not legit - they usually do work if you print them out correctly. I helped US citizens who were already overseas, therefore they were unable to obtain from AAA -- ie, arrived in Poland and were stuck, unable to rent a car. Had them apply on
https://e-ita.org (although once you have the template, you can easily edit the PDF and make for anybody). Just the most important is that you have to follow instructions, and print them out on thick paper, and make it into a booklet with saddle stitching, that it looks authentic. see:
https://e-ita.org/how-to-print-your-international-driving-document-purchased/