One more place?

Not an answer to your question, but why not stop and visit Orvieto on your way between Florence and Rome, rather than a day trip? I hate backtracking.

You mean Varenna, on Lake Como? I'm not sure anybody would characterize it as similar to Florence or Tuscany. But its out of the way. You need to consider how much time you eat up traveling between these places.

Thirteen days is 12 nights?

I'd probably make it simple:

Venice (3 nights)
Florence (3 nights)
Rome (5 nights)

You could stop for a single night somewhere else like Bologna or Verona. Bologna is just a stop off the train; Verona is a little detour.

Some would say just add the night to Florence so you have more time there (given that two days are already devoted to day trips).

Varenna and Lake Como are nothing like Florence or the rest of Tuscany! But your trip is already packed full anyway.
Florence requires two full days which equals three nights without counting day trips.. Too many day trips planned. Why not also stay in Siena, rent a car and explore some of Tuscany.
Rome- a day trip to Orvieto is OK but better to visit it as you travel past it to Rome. Rome to Pompeii for a day is a big trip but can be done. It would be better to save it for a trip to southern Italy.
So..
Venice 3
Florence 3
Siena or another base in Tuscany with car, 3
Rome 4

You have a lot of day trips planned that take time away from the cities you are staying in. I would not recommend adding another place for an overnight. It is a good idea to stop in Orvieto on the way to Rome and give it a couple of hours plus lunch.

Sorry, misspelled
Verona

We will have 13 nights arrive Sunday the 28th of April and Leave Saturday May 11th

Is there somewhere to store our luggage in Orvieto since we will be taking the train? If so that's a great idea. I'm not sure about renting a car and driving in Tuscany. If we had more days in the hill country we would consider it.

Daytrip from Rome to Orvieto works better than on the way down. I don't consider it backtracking since it's so short. Leave Termini around 10:30, you're there before noon. Wander, have a nice lunch, admire the cathedral, shop a bit, catch a 5PM train back and you're back at Termini and Roma in time for dinner.

But Laurel's right--you have a lot of daytrips planned. I'd cut at least one of those but keep this one.

Sounds like you have thought this out pretty well. I would suggest the extra day in Tuscany. I would rent a car in Florence (dropping off in Rome) and visit Siena/San Gimignano for one day, maybe Cortona/Montapulciano for a 2nd day and then Orvieto on the 3rd day. You would need to stay in Tuscany then. We loved Villa Nottola outside Montapulciano, great location to get around Tuscany and wonderful villa with its own restaurant and winery. Trains around Tuscany can be tough as stations may not be near tthe town, car travel is easy.
If you do not want to rent a car, another suggestion would be a day trip from Florence to Lucca and Pisa. They are close to each other and easily accessable by train.

With 2 day trips planned from both Florence and Rome, you'll want 4 nights in Florence and at least 4 nights (preferably 5) in Rome. That doesn't really leave time for another place. The one way it would work is to spend 1 or 2 nights in Orvieto on your way to Rome instead of going as a day trip. Orvieto definitely meets your criteria of different from Venice, Florence and Rome, and it has lots to do and see.

There is luggage storage behind the train station at the info point.

https://www.orvietoviva.com/en/orvieto-train-station/

Do note the closing time.