You've got the right idea about the MSAR's national median, but the problem with the school medians is a bit different. The issue with calculating a median MCAT for applicants *accepted* to a school is that schools tend to offer acceptances to higher stat applicants more often, and on top of that, schools accept more applicants than actually enter their classes by a large margin. Consider the case where an applicant with an MCAT score in the 40's gets in to 10 schools. This applicant only attends one in the end, but the MSAR calculating the median *accepted* MCAT score by including that applicant's score for *all 10 schools*. In reality, especially since higher stat applicants are more likely to have more acceptances (but still only attend one school in the end) and thus their scores contribute to more schools' accepted-MCAT-medians, these medians in the MSAR are (typically) skewed above the schools' *matriculated* applicant medians. The school essentially *is* its students, and if the school accepts a bunch of high stat applicants but most of them go elsewhere, then while their accepted median MCAT may be a 37, the median score for students actually attending may be 34. For applicants this means that evaluating your score against the median for the school's students (i.e. the matriculated MCAT median) is arguably more meaningful than looking at the median for all applicants the school accepted.

Edit: For NYU, the accepted medians are 3.9 and 36, but the matriculated medians are 3.83 and 35.