No, sorry, it is not well-defined. But I should have been clearer. What is not well-defined? Well, the game you are playing. And, without a game, what mathematical question are you even asking?

You cannot just "ignore" or "condition out" the case that there is a car behind the opened door, the game doesn't make any sense anymore then, and what you are measuring then makes no sense anymore with respect to the game. In order to make it well-defined, you need to answer the question what happens in the game when the door with the car is opened.

You can for example play the following game: The contestant picks a door, the host opens one of the other doors, and now the contestant can pick again one of the three doors. If there is a car behind the door the contestant picks, the contestant wins. Note that in this game, the contestant may very well pick the open door. The strategy is now to obviously pick the open door if there is a car behind it, and switch doors if it is not. I am pretty sure, when you simulate this game, you will see that it doesn't matter if the host knows where the car is (and uses this knowledge in an adversarial manner), or not.

The game you seem to want to play instead goes as follows: If the door with the car is opened, the game stops, and nobody wins or loses. Let's call this outcome a draw, and forget about how many times we had a draw in our stats. But you can see now that this is an entirely different game, and it is not strange that the resulting stats are different than for the original game.