BIMBI GRATIS Child Free in Trenitalia for Milan to Venice - age

I think you might have a translation problem. It could be "14 and under" morphed into "under 14". Trenitalia clearly states "under 15". Also the ticket purchase site shows when buying base fare child tickets, "50% discount children 4-15 (not yet completed)".
I'd stick with what you got, even it means having an argument with the conductor. Print out the web page (in Italian) and use it for back-up.

I would definitely get something in writing. We also booked on Trenitalia with a BIMBI GRATIS ticket, but my kiddo was well under the 14 cap. We had to show passport to justify the ticket once on board.

I just remember how I bought a ticket on the Leonardo Express from FCO to Termini. Even though a Rome info website said children between ages 4 and 12 travel free with each paying adults, the ticket inspector told me their policy was "under 12" and not including 12, and I almost got a massive (160 Є) fine. The ticket checker said he was giving me grace because he thought I misread the fine print, when I never saw any age notifications when purchasing that ticket at all! To me, it should say "through 11," or some derivative thereof.

I still think you are OK. The poster with the issue on the Leonardo Express clearly misinterpreted "children between the ages of 4 and 12 travel free with every one fare-paying adult". I would interpret that as being from the 4th birthday to the 12th birthday, so a 12 y-o would not qualify. The regional train explanation for child tickets states "50% discount 4-12 years (not yet completed)", but rules do vary by region. further down it states this:

Lazio - Fiumicino Airport fare 1/Ter : children up to the age of four travel free of charge provided they are accompanied by an adult in possession of a valid travel document; children between the ages of 4 and 12 years old travel free provided they are accompanied by an adult in possession of a valid travel document; in this case, a free ticket for children is possible for each paying adult.

@enbateau, Glad you dodged that 160 EUR fine, but I think you got it wrong.

I also think you are ok. The Trenitalia website has the 15 year age here:
https://www.trenitalia.com/it/informazioni/la_guida_del_viaggiatore/offerte_commercialiperitreninazionali.html#11.html

Scroll down to where it states ” la gratuità del viaggio per i ragazzi fino a 15 anni (non compiuti),”

I checked the Trenitalia website in Italian my ticket ( after putting my confirmation number and generate the PDF in Italian) Bimbi Gratis comes as "Ragazzi (fino a 14 anni compiuti) gratis per gruppi da 2 a 5 persone con
almeno un adulto. Un solo cambio e rimborso parziale fino alla partenza. Soggetta alimitazione posti." - which according to google translation means "up to 14 years old" , so it should include 14 years .
So I am going to discard those English PDF that they emailed and take print out in Italian .

After reading Laura's reply I am more confused .. ragazzi fino a 15 anni (non compiuti)
Ticket PDF - Ragazzi (fino a 14 anni compiuti) .. Why couldn't write the same thing they say in the BIMBI GRATIS promotion only Trenitalia can say if these two things same.
Not sure why the Trenitalia doesn't reply , I have contacted them twice giving all my details through the online Form to contact them , but no reply .

Google Translate, Hopefully Dario, lachera, or Roberto can contribute a better translation
Italian: Ragazzi fino a 15 anni (non compiuti) English: Children up to 15 years (not completed)
Italian: Ragazzi (fino a 14 anni compiuti) English: Children (up to including 14 years old)

I take these as 2 ways of saying the same thing, "Up to the 15th birthday not yet completed" and "Up and including age 14".
Another maddening Italian conundrum. But I still say your tickets are fine.