Pretty simple Amendment by text (only Sections 1 and 5 bear on this):

Section 1.
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. No State shall make or enforce any law which shall abridge the privileges or immunities of citizens of the United States; nor shall any State deprive any person of life, liberty, or property, without due process of law; nor deny to any person within its jurisdiction the equal protection of the laws.

and

Section 5.
The Congress shall have power to enforce, by appropriate legislation, the provisions of this article.

Now let's look at the actual history of this act.  First Senator Howard, one of the authors, at the time of its debate and passage:

So the Senator explicitly stated that the intent excluded those born to foreigners -- aliens -- along with those born to Ambassadors, Consulate officials and similar.

You'd think that would be the end of it.  But it isn't, because we don't read history anymore.  The logic here is quite-clear; a person born to parents who are not Americans are in fact subject to the jurisdiction of a foreign nation.  It belies reason to believe that one is subject to the jurisdiction simply by being present without lineage or other attachment -- I am not subject to the jurisdiction of Canada should I travel there, but I am still required to follow their laws.

Or am I?

Well, that got examined in Wong Kim Ark (United States .v. Wong Kim Ark March 28, 1898) by the Supreme Court.

He was born to parents of Chinese descent who at the time were here legally with permanent residence but due to the laws of the time could not be naturalized.  He traveled with them to China at age 21 and on attempted return to the United States was denied entry as a non-citizen.

Here are the admitted facts by both sides of the case:

'That the said Wong Kim Ark was born in the year 1873, at No. 751 Sacramento street, in the city and county of San Francisco, state of California, United States of America, and that his mother and father were persons of Chinese descent, and subjects of the emperor of China, and that said Wong Kim Ark was and is a laborer.

'That at the time of his said birth his mother and father were domiciled residents of the United States, and had established and enjoyed a permanent domicile and residence therein, at said city and county of San Francisco, state aforesaid.

'That said mother and father of said Wong Kim Ark continued to reside and remain in the United States until the year 1890, when they departed for China.

'That during all the time of their said residence in the United States, as domiciled residents therein, the said mother and father of said Wong Kim Ark were engaged in the prosecution of business, and were never engaged in any diplomatic or official capacity under the emperor of China.

'That ever since the birth of said Wong Kim Ark, at the time and place hereinbefore stated and stipulated, he has had but one residence, to wit, a residence in said state of California, in the United States of America, and that he has never changed or lost said residence or gained or acquired another residence, and there resided claiming to be a citizen of the United States.

'That in the year 1890 the said Wong Kim Ark departed for China, upon a temporary visit, and with the intention of returning to the United States, and did return thereto on July 26, 1890, on the steampship Gaelic, and was permitted to enter the United States by the collector of customs, upon the sole ground that he was a native-born citizen of the United States.

'That, after his said return, the said Wong Kim Ark remained in the United States, claiming to be a citizen thereof, until the year 1894, when he again departed for China upon a temporary visit, and with the intention of returning to the United States, and did return thereto in the month of August, 1895, and applied to the collector of customs to be permitted to land; and that such application was denied upon the sole ground that said Wong Kim Ark was not a citizen of the United States.

'That said Wong Kim Ark has not, either by himself or his parents acting for him, ever renounced his allegiance to the United States, and that he has never done or committed any act or thing to exclude him therefrom.'

The court held that, under the 14th Amendment, he was a citizen. 

Note the distinctions.

• His parents were both domiciled residents.  Under the law of the time they could not be naturalized and while they had not renounced their Chinese citizenship (which would not have been recognized as generally nations will not allow you to be "stateless") they were lawful permanent residents in the United States at all times relevant including prior to Wong's conception and birth.
  
  
• Neither his parents or he committed any act of offense against the United States that would exclude any of them upon legal grounds from said presence in the United States.

Note the actual decision and holding cites history prior to America:

II. The fundamental principle of the common law with regard to English nationality was birth within the allegiance—also called 'ligealty,' 'obedience,' 'faith,' or 'power'—of the king. The principle embraced all persons born within the king's allegiance, and subject to his protection. Such allegiance and protection were mutual,—as expressed in the maxim, 'Protectio trahit subjectionem, et subjectio protectionem,'—and were not restricted to natural-born subjects and naturalized subjects, or to those who had taken an oath of allegiance; but were predicable of aliens in amity, so long as they were within the kingdom. Children, born in England, of such aliens, were therefore natural-born subjects. But the children, born within the realm, of foreign ambassadors, or the children of alien enemies, born during and within their hostile occupation of part of the king's dominions, were not natural-born subjects, because not born within the allegiance, the obedience, or the power, or, as would be said at this day, within the jurisdiction, of the king.

and....

Mr. Dicey, in his careful and thoughtful Digest of the Law of England with Reference to the Conflict of Laws, published in 1896, states the following propositions, his principal rules being printed below in italics: "British subject' means any person who owes permanent allegiance to the crown. 'Permanent' allegiance is used to distinguish the allegiance of a British subject from the allegiance of an alien, who, because he is within the British dominions, owes 'temporary' allegiance to the crown.

There are several other examples cited, all along the same premise and all reaching the same conclusion.

Here is the problem: A person here unlawfully is by definition not within obedience or allegiance to the government as their presence here is not legal in the first instance.  In other words by their own deliberate acts they claim to be beyond the jurisdiction of the United States!

They are exactly equivalent in law to an invading army member who has crossed into the nation in violation of the border with intent to break the laws of the nation.  Such a person is not "subject to the jurisdiction thereof" that a tourist or other lawful entrant is since by their own hand they have acted to deny the right of the government to exert jurisdiction over them and thus DENY said jurisdiction exists.

A lawful permanent resident, on the other hand, has accepted jurisdiction over their person.

Thus the Constitutional Amendment 14 and law stood and Congress recognized this fact (Section 5 power) in 1924 when it passed the Indian Citizenship Act which granted citizenship to those Native Americans who formerly did not have a right to it despite being born on American soil because their parents were in fact "not bound" to the jurisdiction of America as the Tribes had sovereign status.  That they were born on American soil was insufficient under Wong Kim Ark until that act was passed.

Under Wong Kim Ark a person born in America to lawful permanent residents, even thought they are not citizens, is a citizen by birth.  I've argued such a person is not a natural born citizen and thus ineligible to be President (or Vice-President) but there is no other impairment of any sort.  Such a person can hold any elected or appointed office except the Presidency and Vice-Presidency.  For those two offices, and only those two, both your parents must have been citizens at the time of your birth and there is nothing you can do later on to change that.

But a person born to someone who is not here lawfully -- that is, said person is an illegal alien whether their presence and said birth is by subterfuge, force, overstay of otherwise-lawful admission or otherwise is born to someone who has explicitly and intentionally, by their actions, rejected the jurisdiction of the United States over their person and thus, under the original intent and interpretation of the 14th Amendment as decided in Wong Kim Ark is not a citizen of the United States.

And yes, this prohibition also extends to "birth tourism" where the person falsely, in any material respect, declared their status of pregnancy and/or intent upon entry to the United States or where admission is for purposes of employment, tourism, study or otherwise.  Unless said admission specifically contemplated the potential (or fact, as the case may be if you're already pregnant) of childbearing such a child, born to someone without lawful permanent residence status and not a citizen themselves, is not a citizen.

This is clear, simply by the reading of the record, and thus the Trump order is in conformance with it rather than attempting to redefine it.  Note that Trump's order actually is narrower than Wong Kim Ark and original debate on the 14th Amendment allows in that only one of the two natural parents need be lawful permanent residents or a citizen and further, although the order could reasonably retroactively declare this requirement to be the case Trump has deliberately not done so.

Further, the ACLU's "test plaintiff" has no standing because she has no right to be here and generally you cannot, in our legal system which is predicated on common law, bring a civil case where the events leading to your complaint exist as a result of your fraudulent act.  But for her illegal presence the events would not exist and her unlawful action in fact created the circumstance under which she now (as the ACLU's "client") complains.  This demands an immediate dismissal however it is reasonable for the Supremes to take the case anyway because it then sets precedent and puts a stop to this crap on a summary and permanent, absent a further Constitutional Amendment, basis.

I give decent odds the Supreme Court agrees this is entirely within the remit of the 14th Amendment and precedent set by prior Supreme Court decisions since, specifically Wong Kim Ark.