Every other reference is about "citizens" and not specifically "natural-born citizens."
Vattel clearly discusses natural-born citizens, and naturalized citizens. Do you think so little of Vattel's ability to construct a clear argument that you imagine he intended to create a third category category: citizens by nature at birth who are not natural born citizens?
Alternately you can see that Vattel clearly describes two categories, those who are citizens by nature at birth, and those who are citizens by an act of law (or the sovereign). He describes the simplest case of NBC, and from that draws the principle that he applies consistently not just there, to the case that the birthers like to take out of context, but throughout the passage, namely that presumption that the child's loyalty will follow the father's that shows us that the father's rights regarding citizenship naturally apply to the child as well. That's the principle that he applies in the simple case of NBC, note that the place of birth isn't part of that argument.
So when he gets to the more problematic cases, he applies the exact same argument that he applied to the simple case, arguing more than once that it's not the place of birth that matters, but the father's citizenship and loyalty. He's completely consistent about this throughout. And although he doesn't repeat "natural-born citizen" he clearly says that he's talking about citizenship that is acquired naturally at birth.
I suggest that he talks about natural-born citizens, and children who become citizens by nature at birth, applying the exact same argument throughout that discussion, because it's all the same category. Citizenship acquired naturally, at birth, as opposed to citizenship acquired by naturalization (which he *clearly* isn't talking about when he considers these cases of not being born on the home soil).
You can look at it and instead imagine that this man who writes careful, methodical and very clear arguments intended to describe the categories of natural born citizen and naturalized citizen, and a third category of children who are citizens naturally when born but not natural born citizens. But can you apply such a clownish interpretation and keep a straight face?
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