Kathy, I had the same question before our trip last September, traveling with our extended family in a group of 8. I pointed out the difference between the two situations as you just did, and no one ever resolved the dilemma. I thought Dario could but that was when he stopped posting.
I concluded that the distinction must be between Regionale tickets purchased on the website (which can be printed at home or displayed on a phone) and tickets purchased through the app (which require the “check-in” before boarding).
We traveled with no issues on our Regionale tickets I bought on the website, using either the paper copies or mobile displays (I gave everyone on the trip both). We did not have to validate or “check in”.
But that was September, and I believe there was some change in the process in October, mandating the “check-in” for Regionale tickets. Most explanations I see online say that means one must “check in” on the day of travel for all Regionale tickets by validating the ticket in the app, using a link sent to the user.
I am wondering, like you, if this still only applies to the “digital regional ticket’ bought through the app, not the tickets bought on the website. The language in the links you posted above logically supports that interpretation. One way to look at this is that the app tickets are “anytime” tickets that need to be validated before use, just like a paper ticket purchased from a machine or agent at the train station. The tickets bought on the website, on the other hand, whether printed out or downloaded onto the phone, are for a specific train (or one within a few hours of the original departure time). How would one even “check in” with such a ticket? Does the Trenitalia app send you a link to do that, even though the ticket was not purchased through the app? If not, how can they enforce that “check-in” requirement? What about people who do not even own a phone? What are they supposed to do?
I hope we get some official clarification on this. The strict interpretation (one must check in for ALL regional tickets, whether purchased through the website or the app) means that everyone is required to have a mobile phone and use the Trenitalia app. That makes no sense to me. And the language in the upper link you posted above does not support that interpretation.