The 10.10 train out of Termini would mean the 9.23 from the airport with 15 minutes to change at Termini, possibly across many tracks. If you book a Super Economy fare, the ticket is lost if you miss the train, so consider how much risk you're willing to take.
As of this past Aug. 5, a Regionale ticket residing on the Trenitalia mobile site or app requires check-in when you're sure which train you'll take. Here's information from someone knowledgeable on another site:
NEW RULES FOR TRAVEL ON A REGIONALE TRAIN WITH AN ELECTRONIC TICKET (now called "Biglietto Digitale Regionale" or "BDR"):
1. These tickets are purchased from the Trenitalia website or app using a credit or debit card;
2. You must select a specific regionale train for a specific time and day, and are not free to board any other train;
3. Prior to 23:59 (Italy time) of the day prior to your selected travel date, you may change the departure time or day of your regionale train;
4. On the day of travel, you may still change the departure TIME to a different train, but cannot do so after you have "checked in" to your originally selected train;
5. On your travel day, and prior to boarding your train, you MUST "check in" to your selected train online OR by hyperlinking to the "check in" tab on the ticket confirmation you received by e-mail or SMS when you purchased the ticket online or through the app. If you don't "check in" you will be deemed to be traveling without a valid ticket and risk being fined!!! Once you "check in" the electronic record of your selected train will reflect a "convalidato" script and a green bar below to reflect that the ticket is now valid for use.
6. Make sure that the electronic record of your ticket (on your cell phone, tablet or PC), or a paper copy thereof, has a scannable QR Code as well as the green validation bar reflecting that you have "checked in", as you MAY be asked to show this by the ticket controller on the train. You should also have picture identification with your name which must match the name on the electronic record of the ticket.
Rules in Italian:
https://www.trenitalia.com/it/treni_...regionale.html
So this would probably apply to the FCO-Termini and Florence-Siena segments; for the Rome-Florence segment you would already be reserved on the train you've chosen; no problem as long as you make that train.