@capecruzer
The question is what do you want to do in Pisa? If you take a taxi and get there by 9, do you want to be in Pisa for 4 hours? Maybe you want to climb the tower, go to the Cathedral, Baptistry and Camposanto and use the full 4 hours. Maybe you only want to see the tower. If it is the latter (with hopefully a look at the cathedral), you can be done in less than two hours. Then you would like to take an earlier train to Vernazza around 11 to 11:30. After some time in Vernazza you could take the ferry or train to Monterosso, and later a train back to Livorno.
You can buy tickets for any of the trains at the Pisa station. It might not be a good idea to buy the Pisa-CT ticket ahead of time because it would not give you any flexibility.