They never define what natural born citizen means!
They didn't define the word "arms" either, because they figured people wouldn't be dumb enough to forget what it meant.
But I can offer some insight on this issue. We Americans are familiar with the word "citizen" only because we adopted it back in 1776.
It was an uncommon word prior to 1776, and it didn't mean the same thing as we take it to mean today. In English, the word "citizen" meant "denizen of a city". It meant someone who lived in a city.
There was only one place in the world where that word was used to refer to the inhabitants of a Nation, and it wasn't England.
The US usage of the word "citizen" comes from Switzerland. That is the only nation which used the word to mean inhabitants of a country, and they did so because they were created by the amalgamation of different states into the old Swiss Republic.
Their "Priest charter" of 1370, (considered the foundation of the old Swiss Republic") specifically mentions "citizens of cities, and citizens of "the country. "
We got the word from the Swiss, and therefore we got the meaning of the word from them as well. The British word for "inhabitant of a nation" was "subject."
We picked the word "citizen" to demonstrate that we were founded on different natural law principles than those which gave us the concept of "subject."