Obviously this debate is reignited by the Cruz candidacy so here’s how I like to look at things when its not clear from plain reading (in this case I think it is clear and you explained it well). If one is to allege that Cruz is not natural born, than what is he? Naturalized? But in order to be naturalized, one has to take affirmative steps to accept it. For example if a Canadian came here, he would have to ask for citizenship or in the case of a child have his parents apply for him. He has to forswear allegiances to other countries and take a loyalty oath to America, and be sworn in by a US official. How can one be neither naturalized nor natural born? It’s impossible. It has to be one or the other. So in this case “Natural born” means acquired at birth. It doesn’t matter what soil the birth occurred upon, if he is born a citizen he is ‘naturally born’.
Of course this doesn’t mean other things can’t also be true. For example many countries including the US will consider someone born in their country a citizen too. The USSC has ruled on this issue, saying that citizenship is a matter for each country to decide on their own. There is nothing the US nor even the person himself can do about it if that country chooses to record you as a citizen. If your parents are/were both stationed in a foreign land in a military or diplomatic assignment and you happened to be born outside the US borders, that country could claim you even if you have no desire to accept citizenship. This is important because of the question of the opposite: if they did not claim you and the US law was that you would have to be naturalized, then you would live without citizenship to any country. Not really possible, except in theory.
So if Cruz did not have to apply for citizenship, did not have to take an oath, his parents were American and recorded his birth to appropriate US government agency as being born from a US citizen as an America (e.g. he is ‘naturally born’), then he is a natural born citizen no matter where the birth happened to take place.
That's the fallacy, in a nutshell.
There are several statutes that convert a person who otherwise would not be a citizen, into a citizen, merely by the circumstances of birth meeting the conditions stated in the statute.
In order to be naturalized, one's citizenship must depend on an Act of Congress. That's awkward phrasing, but that's a form of definition for "naturalize."