I just rented a vehicle from an NPC and I don't see the option to enter the cockpit under the "Personal Data Assisstant" title. I don't even see that little stick man. And yes, I'm in the right spot on the grid.
It should be something along the lines of:
Position -> Enter -> Select Vehicle
Once you're in the vehicle, you should be able to enter the cockpit from the new link that appears on your right hand menu. You may have to move to the cockpit room first, if the vehicle has more than one room.
"May the Grace of Ara go with you, and His Vengeance be wrought upon your enemies."