...foreigners, aliens, who belong to the families of ambassadors or foreign ministers accredited to the government of the United States but will include every other class of personsThese are the standard and longstanding exceptions both for "natural born subject" and "natural born citizen."
You also may have missed this from Senator Trumbull, chairman of Judiciary,
The senator from Missouri and myself desire to arrive at the same point precisely, and that is to make citizens of everybody born in the United States who owe allegiance to the United States. We cannot make a citizen of the child of a foreign minister who is temporarily residing here. There is a difficulty in framing the amendment so as to make citizens of all the people born in the United States, and who owe allegiance to it. I thought that might, perhaps, be the best form in which to put the amendment at one time, 'that all persons born in the United States, and owing allegiance thereto, are hereby declared to be citizens;' but, upon investigation, it was found that a sort of allegiance was due to the country from persons temporarily residing in it whom we would have no right to make citizens [p572 of the Congression Globe on the debates over the fourteenth amendment]
To break it down for you: