Your history teacher is incorrect. But it is a common misperception. The Constitution refers to "natural born citizen" not "native born citizen" as some history texts report it.
I don't understand the difference. Anyway the Panama Canal Zone was U.S. jurisdiction at the time. I went to college with a kid who was born on a U.S. Army installation in Germany. Isn't that considered the same as being born on American soil?
I think it is a little bit more complicated than that. I dont think you can interpret "natural born" in any way other than "native born" without running into complications. Like the US, many countries accept any one born in their country (irrespective of the citizenship of their parents) as a Citizen of their country. It is only at the time of majority (18) that you choose which country is your homeland.
From wikipedia:
One side of the argument interprets the Constitution as meaning that a person either is born in the United States or is a naturalized citizen. According to this view, in order to be a "natural born citizen," a person must be born in the United States; otherwise, he is a citizen "by law" and is therefore "naturalized."[5] Current State Department policy reads: "Despite widespread popular belief, U.S. military installations abroad and U.S. diplomatic or consular facilities are not part of the United States within the meaning of the 14th Amendment. A child born on the premises of such a facility is not subject to the jurisdiction of the United States and does not acquire U.S. citizenship by reason of birth."[6]