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Doesn't that contradict Vattel's definition?

Yes, of "natural born citizen" at least, except in the case of children born to military or diplomats serving the nation, but outside of it. It may have been an attempt to extend just plain old citizenship to such persons, and later they realized they had overstepped in trying to define a Constitutional term. In fact in 1795 they dropped the "Natural Born" language when they repealed the 1790 and replaced it. The language is thus no longer in effect and has never appeared again in the statutes. The relevant part of 1795 Naturalization act follows.

(It's in Section 3, on the next page.)

...; and the children of citizens of the United States, born out of the limits and juridiction of the United States, shall be considered as citizens of the United States: Provided, Theat the right of citizenship shall not descent to persons whose fathers have never been resident in the United States.

Vattell however would have such persons be born citizens:

From Law of Nations, Book I, section 215:

It is asked whether the children born of citizens in a foreign country are citizens? The laws have decided this question in several countries, and their regulations must be followed. By the law of nature alone, children follow the condition of their fathers, and enter into all their rights (§ 212); the place of birth produces no change in this particular, and cannot, of itself, furnish any reason for taking from a child what nature has given him; I say "of itself," for, civil or political laws may, for particular reasons, ordain otherwise. But I suppose that the father has not entirely quitted his country in order to settle elsewhere. If he has fixed his abode in a foreign country, he is become a member of another society, at least as a perpetual inhabitant; and his children will be members of it also.

Of course BHO Sr, did no such thing, He did not "quit his country", but rather returned to it after completing his educatoinal sojurn in the US.