My understanding is that a trips start and end with Ignition On.
That is nearly correct. The trip start and end on when the vehicle actually starts moving. This is better represented by the figure below:
More information can be found in the Product Guide in section 4.Trips.
Now, If a driver is assigned to a vehicle while the vehicle is driving (during Section B above), the driver will be assigned for the current trip. If the driver is assigned to a vehicle during a stop (section C or D), the driver will be assigned to the vehicle for the next trip.
What happens when there is a driver change while the vehicle is on? Is a new trip created? If not, which driver is assigned to the trip?
No new trip is created. If a driver is assigned while the vehicle is moving, that will override previous drivers. So, the most recent driver assigned will be assigned to that trip.