If you are born within the United States of America, you are a natural born citizen with all the rights that entitles you to (such as Medicaid, food stamps, etc.). This is an important issue in border communities. Some people come into Texas simply to have children, then return to Mexico. The children are then citizens of the USA and can assert their citizenship rights. I believe, however, that they must return to the United States before their twenty-first birthday, or they lose their American citizenship. The United States does not recognize dual citizenship, so a person must choose USA or Mexico (or whatever) by age 21.
Of course, not being an immigration lawyer, I could be wrong. I just know that it’s been an issue. Hospital emergency rooms receiving Federal money (such as Medicare and Medicaid) cannot refuse service to anyone with dire medical need or any pregnant woman in labor. Therefore, when a pregnant legal or illegal alien presents herself in a Texas emergency room, they must (and will) give her maternity services. Personally, while some people think that this is wrong, I think that it is an appropriate homage to the fact that ours is an immigrant nation.
As for the OP, there’s an extensive scholarly paper on this exact question here.
It appears that the primary reason for the “natural born citizen” clause was to prevent a foreign power from having undue influence over the presidency.