2 weeks italy. Venice, Florence, Chiante region, Cinque Terre, and Rome

Probably easier to do the 2nd plan and drop the car off in La Spieza.
From there you can easily take the local trains to whichever village in the Cinque Terre you are staying in.

I imagine that would be easier than dropping off outside of Rome and getting from your drop off location to your hotel in Rome by public transit. I know some drop off / pick up in Orvieto but they also usually have Orvieto as part of their plans for a few hours or an overnight stay. Trains to Rome from Orvieto will be frequent and a short ride.

I agree, 2nd is best

I agree with the others, Option 2 is preferred. The trip from Venice to Florence will be via high speed train which travels at up to 300 km/h, so much more efficient than by car. For your visit to the Cinque Terre, drop the car in La Spezia and then take the local trains to whichever of the five towns you'll be staying in.

As this is your first trip to Italy, note that there are some potentially expensive "caveats" for both train and car travel. A brief summary....

• Each driver listed on the rental form will require the compulsory International Driver's Permit, which is used in conjunction with your home D.L. These are easily obtained for a small fee at any AAA office, and are valid for one year.
• You'll have to be extremely vigilant to avoid the dread ZTL (limited traffic) areas in each town you visit. Do NOT drive in Florence, as the city is just about saturated with automated ZTL cameras, and EACH pass through one will result in an expensive ticket!
• You'll also need to be aware of automated speed cameras, including the devious Traffic Tutor system on the motorways, which registers both instantaneous speed as well as average between two points. Violate either parameter and expensive tickets will follow.
• With the fast trains such as the Freccia, tickets are specific to train, date and departure time, and can ONLY be used on the one train listed on the ticket. If you board another train by mistake, you'll be subject to hefty fines which will be collected on the spot.
• With slower Regionale trains, locally purchased tickets MUST be validated (time & date stamp) prior to boarding on the day of travel. Failure to do that will also result in hefty fines collected on the spot.
• When riding Vaporetti (water boats) in Venice, tickets must also be validated by holding against the electronic reader. If you get a beep and a green light, you're good to go. Again, hefty fines if you forget.

When is this trip taking place?

Definitely no. 2.
Venice (VE)>train from VE to Florence>drive to Chianti > drive to La Spezia (SP)/return car> train from SP to 5Terre > train from 5Terre to Rome.