The original phrasing of the problem is not ambiguous at all:
> 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?
You can change the rules however you like but then it’s no longer the Monty Hall problem, and that’s what we’re discussing: a brain teaser with well defined rules and probabilistic interpretation.
That's not the original phrasing because that's from 1990, have a look at the wiki page you linked. The original problem does not specify that Monty will never open a box with the keys in. The game would still work if this was the case as Monty offers to buy the box before doing so. In the original problem, Monty says the chance is 1/2, which would actually be true if they had opened them randomly. (edit - a game where the host offers to buy the box you have before another one is opened is essentially the end part of deal or no deal).
Even the 1990 version however is ambiguous when it comes to puzzles. It just says they open a door which has a goat.
The solution originally posted explicitly assumes it to be the case, which is fine.
> 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?
https://en.wikipedia.org/wiki/Monty_Hall_problem
You can change the rules however you like but then it’s no longer the Monty Hall problem, and that’s what we’re discussing: a brain teaser with well defined rules and probabilistic interpretation.