The use of sets really clarifies things. Set B has a 2/3 chance of containing the car-hiding door. A fair coin or an RNG chooses which door in Set B to open. If the car is revealed, the game is over and you don't have an opportunity to switch. If a goat is revealed, Set B still has its 2/3 chance of containing the car-hiding door so you should switch to the remaining door in Set B.

Note that this depends on whether Monty's choice of door was informed by the location of the prize

That's part of the setup. If it's not, it's not being told correctly. It's usually in the form of "...and Monty opens a door that is always a goat..."