I think I screwed up our train reservations

If it came up with a price, both of your names were shown, and you have assigned seat numbers, you are fine. Through the weirdness of Trenitalia fares, sometimes Bimbi Gratis costs more because there is some flexibility in re-booking, as I recall. But if you have the seats you are good-to-go. The capotreno rarely checks beyond butts-in-seats.

Right, you have valid tickets for the legs already booked, sounds like you got a reasonable (nonchangeable) price, and you don't need to do anything more. The Bimbi Gratis offer only applies if the adult pays full fare but other (advance) discounts may be a better deal.

Do book him again as a child for the next purchase, unless it's for a regional train; those have a lower age range and are more restrictive when booked online, with no advance-purchase discount.

Thanks! I am looking at this page on Tentialia: http://www.trenitalia.com/tcom-en/Offers/Bimbi-Gratis

Which I find a little confusing. But it sort of reads to me like I was supposed to choose the Bimbi gratis option if I want him to travel free. Since the trentalia website doesn't have a place to enter his age, I'm wondering if what I have actually booked is an adult ticket for me and an under age four ticket for him?

Here's the language from the Trentalia website I linked above:

Bimbi gratis is the offer from Trenitalia dedicated to children under 15 years of age travelling with at least one adult, in groups from 2 to 5 people.

The offer allows children under 15 years of age to travel for free, while the other members of the group pay the Base Price for the selected train and service.

To purchase the Bimbi Gratis offer on this website proceed as follows:
enter the number of adults and children (at least 1 adult and 1 child)
in the page "Choice of travel" select the option "View other prices and services"
in the next page select in the scroller the offer "Bimbi gratis" for all passengers.

The Bimbi Gratis offer holds if you bought a full-fare (base price) ticket. How much did you pay?

Super economy

You already got the best price by booking Super Economy for both. I think the lowest fares are relatively clear to select on the price chart when you enter one adult and one child. Children under 4 don't need tickets at all, so the children they book online are just age 4 and above.

So I paid for one super economy ticket. And nothing for my son. When I compare by looking at how captain train, which does have a place for me to enter my son's age, shows prices it shows me at the full super economy price and my son at a further discounted price. Which makes me think that trentalia is treating him as under four because I selected child and then did not choose the Bimbi gratis offer.

I am just concerned that when I board the train with my son, I will be hit with a fine because the ticket I have shows him as a child.

He is a child. If you want to forward the confirmation email to us, we can take a look: rick@ricksteves.com. But if you have two seat assignments, then you should have two tickets. Children under 4 don't get seat assignments, either; they're expected to sit in your lap, or wherever you find space.

Although my last message about the price chart was from memory, I did a new test now and it's pretty clear to me that the 2nd class Super Economy total fares shown for one adult and one child in 2nd class are either 29 or 34 euros between Rome and Florence, depending on departure time. If I delete the child and refresh the screen, the single adult price is 19 euros. Trenitalia shows the price for the whole group that you requested, not per person.