The citizens are the members of the civil society; bound to this society by certain duties, and subject to its authority, they equally participate in its advantages. The natives, or natural-born citizens, are those born in the country, of parents who are citizens.
From "Law of Nations",Volume I, section 212. Now I know that is the 1793/97 translation. But the founders had the original French version, which many of them could read and translate for themselves, The later translation is just better in this instance. The original French said (less diacritical marks, sorry about that) :
Les citoyens sont les membres de la societe civile : lies a cette societe par certains devoirs et soumis a son autorite, ils participent avec egalite a ses avantages. Les naturels, ou indigenes, sont ceux qui sont nes dans le pays, de parens citoyens”
Now do you see a footnote associated with “natural born citizen” that references vattel Law of Nations”,Volume I, section 212