I agree that this is not as precisely stated as it could be. And unfortunately there's a type of logic puzzle that relies on people understanding these implicit quantifiers and being prepared to ignore them, which sometimes makes it hard to determine if they're supposed to be assumed present or not. But the most common interpretation is to include it.
But the problem described a specific instance. You are reading it wrong. Right now you are just digging deeper into a hole.
If the problem actually was told as you describe then you would be right, but it isn't. It just describes a single instance, you are at the show, the host opens a door, what do you do?
According to Wikipedia, the original problem statement was like this:
"Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?"
Then I think the confusion might stem from the concept of probability. I think what is meant here is probability from the point of view of the deciding person (given their knowledge). That does not have to be the real probability at all.
I don't have a good example and I am not currently well read on it, but I remember from some Bayesian book the example of the lawn being wet. So either it has rained, or the lawn sprinkler had been on. And you can reason about the probabilities. But of course either the one or the other happened (in a simplified logic world). There are no probabilities about it. The probabilities are only the estimates of the person trying to infer if it has rained or not.
OK so strictly speaking, you could create zillions of variants of the Monty Hall Problem from the scarce description. Maybe the host looked at the player funny before the show, so the player raises his estimates for the likelihood of the host being a crook (as in your casino game). Or any number of other things could affect their reasoning.
So it would be merely convention of the mathematical puzzle to assume the simplest case of everything else being random. And then of course there will be puzzles that trick you exactly with that assumption, to teach you that you shouldn't make too many assumptions. OK.
> You are on a game show. There are three doors, one has a prize behind them. You pick a door. The host who knows whats behind the doors opens one of the doors and reveals that it was the wrong door. Should you switch?
This description doesn't contradict my scenario at all. it doesn't say "The host will always reveals a dud no matter what happens".
Now, your intuition tells you that my scenario feels wrong since the host doesn't always do the same thing every single time, but that is how math and statistics works. We only know that the host opened this door this one time, we aren't given information about how the host acts in general. My interpretation is therefore just as valid.
Of course, for the real game the host always opened a dud door. But the original problem description didn't say that, it just described a single instance. In a single instance there isn't enough information to tell. And no, if you assume that the reader must know how the original game show worked then the problem isn't a properly defined problem, and the answer could be anything. That is how math works.