Train connections with limited time window

No, the train does not wait. Moreover it not guaranteed that your train from La Spieza to Pisa will leave on time. Most will tell you the trains are late most of the time.
However that route La Spieza to Pisa Pisa to Florence are served by Regional trains which you don't need a reservation for, meaning if you miss the one you want you can take the next one as long as it is a Regional.

Otherwise there some trains that go directly from La Spieza to Florence depending on the time.

Some train stations are very small, but even with those five minutes is not nearly enough time to get off of one train and find the platform where the next train is located. And then finding the right car to get on. All while hauling luggage...

If you miss a Trenitalia train with reserved seats because another Trenitalia train is late, your connection is always protected. You can get a new seat reservation and a new ticket for free at the counter. These are the rules when you get a through ticket on the official Trenitalia.com site. I don't know what happens when you buy on a re-seller's site.

Note that If you miss a Regionale train for whatever reason, you can freely take the next couple of Regionales. Since there are no reserved seats on Regionale trains you don't need to get a new ticket at the counter.

And then finding the right car to get on.

You can get on the first car and walk inside the train. Difficult but doable even with luggage in tow.

Trains are mass transit. Those short connections are on purpose, as the railways know people like to keep moving, and do not like to wait.
Anyway, relax. The moment you have a train ticket the railways will transport you. Also regional trains do not have reservations and so there is no problem taking a different train than the one you originally intended.

Thanks everyone for the great info!

I know the connection sound crazy but it has always worked out for us. Download the Trenitalia app and sometimes it will show you the track you are going to before you get to get station (saving you a few precious moments looking at a screen). Be at the front of the door and ready to move when the train arrives. We had a group of 10 family that made a few of those and we said if we were to ever form a band we would call it the “Tight Connections”. Humor always helps.

Wait till you come to Switzerland where connections of 5 minutes (or even less) are very common. In fact, from the point of view of Switzerland any time a railway forces you to spend more time than strictly necessary at a station they have failed.