Overlap of usage is a historical phenomenon, a matter of fact, I suppose, and not a fallacy: 2 + 2 = 6. No sir, your answer is wrong!
But you make a good distinction, what they like to call the fact-value distinction.
I think you are mixing posts. I most certainly was not speaking of "mathematical reality" when I said that reality is never incorrect. I do not distinguish between different types of reality. Those distinctions are illusory. I stated that reality is never incorrect, and if it is found to be incorrect then either the method to analyze it is flawed or the observer is artificially constraining his vision (most likely unintentionally).