www.trainitaly.us

I've also looked at www.italiarail.com. They charge a small fee for tickets. I'd check their prices (in USD only) vs what is on the trenitalia website with a current currency conversion rate.
www.trainline.eu also does trenitalia tickets, and they are very reputable.

www.trainitaly.us website says it is in New York and uses the logo of trenitalia. The price is about the same as in the trenitalia website. I looked at the travel agencies in America listed on the trenitalia website but www.trainitaly is not on the list so I wonder if this is a legitimate booking site

Why don't you use the official website www.trenitalia.com?
Do you derive some enjoyment from the possibility of being scammed by some Russian hacker?

This past week I booked tickets through the trenitalia website with no issues. I also booked tickets for the Italo train on its website with no issues. I prefer booking directly for transportation and hotels.

I use trainline.eu for trains in Europe (including here in France). It's so easy and no extra fees.