First, Uncle G, I agree that part of the problem in the article is a disruptive student.
There are fundamentalist groups that would say that there are only two sexes. They will adapt their language if you point out to them that these days, due to multiple meanings of the word sex, we use gender (just as the word "gay" no longer can be used to mean "happy and light-hearted" because it has been adapted for other use).
Where the problem comes in is that gender (in the physiological sense) and gender identity are distinct and can differ. Further, there are cases where children are born with variant chromosome situations or reproductive structures where there is a lack of clarity as to which way the child will go. And it appears that the brain can have more than just two different configurations, which makes one wonder about various brain-body combinations.
regarding physical brain differences including structural differences:
You can search for Klinefelter Syndrome and see behavioral issues as well as physical ones.
The old term "hermaphrodite" comes to mind here. It is a non-scientific term that is yet another physical condition clouding the picture of sexual variation.
How many genders are there? The fact is that two genders are not enough even when you limit yourself to physiology since there IS such a thing as the brain and the body being in disagreement. When the wiring and the plumbing disagree, you have either a homosexual or a bisexual. When that happens, the language requires either some special word (usually taken as a pejorative) or you have to use a descriptive phrase. Therefore I cannot tell you how many genders there really are, but there exists a spectrum of behaviors for which the extremes of the spectrum are "traditional male" and "traditional female."
The articles I posted regarding brain structure represent findings using brain scans on homosexual people. The fact that the brain scans can detect the situation very strongly suggests something else: Brain structure is fixed during gestation; therefore, homosexuals are born that way. (Do I hear Lady Gaga singing in the background?)