ALL trains/trenes by TrenItalia are included. An there are no other trains IN this area-which is quite overhyped and probably also gigantic overvisited this summer. Local trains may however have only 2nd cl., not sure. And even if not-expect that noone respects 1st in these trains.

But fares for short distance are so low in ITalia that it is far cheaper to just pay as spend passdays (cost mostly 30-40€) for this. IF you have a continue pass-and certainly from the promotion, then of course you do not need to.

Tr-It also has a quite cheap 3 day pass for ALL its REGIONAL only trains anywhere in IT for even less as that 47€-but it excludes the faster IC/Frecce trains-that about anyone uses to visit the 5 earth from large cities.

I found this on a website "The Cinque Terre Express is a fast-regional train offered by Trenitalia which covers the route from La Spezia to Levanto and reaches the main cities of the Cinque Terre National Park". So from what you told me, our every day travel interrail cards do include cinque terre? So we will not need other extra cards?

I dont really care about 1st or 2th class haha but thought it might be important information. The plan now is to stay in Vernazza. I know its super touristy. But we have to end in Pisa and this looked the most straight forward route. I cant switch accommodations to much. 

Sure, the Cinque Terre Express is only a marketing thing to sell tickets for this area more expensive that in others, but it is a normal local train that doesn’t stop everywhere (Regionale Veloce - RV - English: Regional Express.) No reservation or fee needed. Only add the trip to your pass and hop on the train.