In truth it actually is a bit confusing. So hopefully I can clarify how it works.
Firstly, if you hire a motorhome (or campervan - same thing, just a different word), they all cross. And pretty much whichever motorhome company you hire off can also book a ticket on Interislander for you and the motorhome.
If you've hired a rental car from one of the international brands - Avis, Budget, Europcar, Hertz or Thrifty - those companies do not allow their vehicles to cross. When you get to Interislander's ferry terminal, it's as if you were at an airport. You need to park in the rental car returns carpark, unpack the car, put the car keys in the key return box and then check in. When you get to the other side, just reclaim your bags off the carousel, proceed to the rental car kiosk to sign the paperwork and pick up a set of keys for your new rental car.
In general the New Zealand-owned rental car companies offer good value rates and allow you to take the vehicle across. You just simply drive on and drive off. Most of them will also book your ferry ticket at the same time as your car booking and if your hire is long enough they often pick up the cost of the vehicle crossing. Check out the websites of Ace, Apex, Ezi-Rent, Jucy or Omega for deals.
And finally a little tip: Interislander's ship, Kaitaki, is the largest of the ferries. Because it is the only ship that loads from the bow at Wellington and discharges from the stern at Picton, there is no need to turn inside the ship; you're already pointing in the right direction to drive straight off once at destination. Too easy!