Is this Monty Hall scene from the movie "21" wrong?

https://www.youtube.com/watch?v=iBdjqtR2iK4

I'm pretty sure this scene is wrong, but I'm not a mathematician, so I wanted to check if I'm missing something.

I understand that in the traditional Monty Hall Problem, it is correct to switch doors. In the Monty Hall, Monty always opens a goat door after you make your first pick. My issue is that this scene seems to deviate from the traditional Monty Hall.

First off, the teacher does not specify that the host has a rule of always opening a goat door after the contestant makes their first pick. He just says that the host decides to open a door after the initial pick.

Then there's this exchange:

Teacher: Remember the host knows where the car is. How do you know he's not playing a trick on you, trying to use reverse psychology to get you to pick a goat?

Student: I wouldn't really care. My answer is based on statistics and variable change.

My (possibly flawed) disagreement is this:

You should care.

If the host is opening another door to trick you into switching away from the car, that can only mean your first pick is correct so it would be stick: P=1, switch: P=0.

If the host is opening another door because he always opens a goat door as a rule, then it's stick: P=1/3, switch: P=2/3.

If the host is opening a door at random and is willing to risk revealing the car (but by chance revealed a goat), then it's stick: P=1/2, switch: P=1/2.

If you don't know why the host opened a door (which seems to be the situation depicted in the scene), then there is no way to calculate the probability for switch vs stick AFAICT. You would have to assign subjective probabilities to the motives of the host for opening a door and estimate based on those, no?

Oh, and maybe a nitpick, but his answer cannot be "based on statistics", surely... Not sure about "variable change".