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I think that the missing "variant" that you are talking about is that each of the doors is an independent probability. So, doing math and saying that you were 66% and are now 50% likely to have the wrong door is incorrect.
Each door is independent. In statistics class, you'll see that there are different formulas for dependent and independent probabilities. In this case, Monty Hall opening the first door for you does not statistically change the percentage of chance that you picked the correct door in the first place.
For instance, if you flip a coin 9 times and get "heads" nine times in a roll, what is the probability of getting "heads" on the tenth time? 50%. The 9 flips beforehand statistically had nothing to do with the 10th flip. It's an independent probability.
That's how I remember it in class, anyway.
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