All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the State wherein they reside.
Let's leave the fanciful interpretation thing to others, shall we?
The only question here is who is subject to the jurisdiction of the United States.
I can't answer for conditions back in 1812, but as of today, my understanding of American jurisdiction relevant to this matter is that it encompasses the area within our National borders, excluding foreign embassies and the like, so any children born of the 9-11 hijackers on our soil would become citizens, and it is that same understanding of jurisdiction which has allowed Zaccharias Moussaoui, a French citizen, to be arrested on our soil and tried in our courts - he has plainly fallen under our jurisdiction.