However, if someone creates a convincing argument that it can only mean one thing, then he or she should share that argument with the American people and if it is sufficiently convincing, the American people and their electors will accept it and employ it in selecting presidents. So far, I think most people seem comfortable with equating natural born citizen with citizenship at birth and I think that's what they've been doing.
See, one of the problems here is that most people see candidates like McCain and Cruz (both of whom spent their lives living in American neighborhoods, studying in American schools) as completely American. They don't see them as strangers to America who have come from some foreign land to subvert our system. So, using those people as examples of what the natural born citizen clause is designed to protect us from is counter-productive. When they see POW-United States Senator McCain, they don't see any great danger and they aren't motivated to narrow the natural born citizen clause in order to exclude his candidacy.
This year it looks like it's going to be Trump vs. Clinton. I guess there will be a few folks who will invent a claim that Trump is really the son of some Arab rug merchant or that Clinton was really born in France, but it won't work any better than it did the last time. Somebody will sell a few books and CD's, but most people won't be fooled. This kind of baloney - "The True History of So and So" - has become part of our electoral process. But, you're not required to take it seriously. You're not required to be a sucker. It doesn't mean anything in the long run.
Point 1. The adjective "natural" in the phrase "natural born Citizen" must not be superfluous and without contributing further significant restrictive meaning to the phrase it modifies, therefore a natural born Citizen must be a significantly restricted subset of born citizen (citizen at birth). Clearly the two phrases (citizen at birth and natural born Citizen) must mean significantly different things with the latter being a restrictive subset of the former. To suggest otherwise is insulting to the deliberate, articulate elegance of the founders who meticulously crafted the Constitution.
Point 2. Three inherent characteristics at birth are generally considered to contribute to citizenship and to one's natural allegiances, these being: the citizenship of one's father, the citizenship of one's mother and the land of one's birth. The founders' explicit stated purpose of restricting the presidency to only natural born Citizens was to provide a strong check to prevent the republic from falling prey to a presidency (and military) subverted by foreign influence and intrigue. It strains all logic and common sense to beyond credulity to suggest that the founders would believe that a strong check would be provided by requiring just one of these characteristics (the least restrictive combination) rather than all three (the most restrictive combination).