There isn't, but original meaning is the dominant branch of originalism, and it does not care about intent. Consider Justice Scalia's words from the man himself:
You will never hear me refer to original intent, because as I say I am first of all a textualist, and secondly an originalist. If you are a textualist, you don't care about the intent, and I don't care if the framers of the Constitution had some secret meaning in mind when they adopted its words.
Speech
On the other hand, with regard to the phrase in question, Scalia, in colloquy, during a 2001 case, said on the record that the intent of the phrase was that the children of an American colonial woman and a British officer who married her during Britain's occupation of the colonies would have a child whose loyalties would be divided and that that was what the Framers wanted to prevent.
See the exchange between Mr. Justice Scalia and Mr. Davis in the transcript of the hearing in the case of Tuan Anh Nguyen v. INS, 533 U.S. 53 (2001).
There was no "hidden intent" involved in taking the phrase from Vattel at the time. It was an intention that was openly known and discussed. It was a part of common discourse of the time that what the phrase meant was born in the in the country of parents both of whom were citizens. You mistake "hidden" intent for what was a defintion accepted in the normal discourse of those involved.