You have asserted that the reason doctors can accurately tell patients their chances after a diagnostic test even if they perform poorly on the word problem is because they are confused about the term "false positive". But the problem can be phrased without using the word "positive" at all, and people will still get it wrong if it's phrased in terms of probabilities and get it right if it's phrased in terms of relative frequencies. So the fact that doctors can tell patients their chances after a diagnostic test even if they perform poorly on the word problem has nothing to do with them being confused about false positives.