My claim is correct, because it was for quantum mechanics.Ah, now I think I see the source of confusion. One should distinguish two things:
1) SINGLE measurement, from which no information about probability distribution can be extracted (except that the obtained value has probability larger than one).
2) Statistical ENSEMBLE of similar measurements, from which the probability distribution can be extraced.
I was talking about the former, while it seems that you are talking about the latter. If I simultaneously measure position and momentum ONLY ONCE, I cannot extract any information about the joint probability distribution.
But then again, even in classical mechanics I can repeat many times the simultaneous measurement of position and momentum. From such a measurement I CAN extract the joint distribution. Moreover, by using the theory called classical STATISTICAL mechanics I can even predict or explain the joint distribution I measured. So your claim that "classical definitions for joint distribution don't exist" is certainly wrong.
What I don't understand is: what do mean by an "accurate" measurement? To define an accurate measurement in some sense, one needs a "correct" answer. In classical mechanics, one way to define a "correct" answer is that one correctly infers the value of the property that the particle had at a certain time. However, for joint measurements this definition of "correct" can't carry over to quantum mechanics, because the joint distribution of position and momentum doesn't exist in general.