No, it does not equate native and natural born citizens as the same. It states that both type of citizens are distinguished from Aliens or Foreigners.
While not quite as precise as mathematical logic, when someone states “it was never doubted that all children born in a country of parents who were its citizens became themselves, upon their birth, citizens also. These were natives, or natural-born citizens, as distinguished from aliens or foreigners.” The verb “were”, past tense of to be, asserts equivalence or ‘the same thing as’. If “Natives” is the same as born in country of citizen parents, and “natural-born citizens” are the same, the two terms are the same by the transitive property of equality, which both languages of mathematics and English accept as axiomatic.
The reason citizens were left out of Justice Waite's statement is that the Constitution only defined one class of citizens. There are two, natural-born, and naturalized. Naturalized citizens are defined by men and our framers properly left those definitions to Congress, saying as much in Article 1 Section 8, telling Congress to create “an Uniform rule for Naturalization.”
Natural-born citizens are citizens by the ‘natural law’ upon which our Declaration and Constitution are based. Some prefer to say citizens by “God”. Either way, it was the predominant philosophy of the Enlightenment, and the basis for the legal philosophy not just of de-Vattel, but of Pufendorf, Bynkershoek, Leibniz, Wolf, Grotius, and dozens of others. Vattel didn't invent it, but wrote more clearly than most, wrote independently of the Vatican or Protestants, and at just the time that our founders were plotting to throw off the tyranny of a monarch they knew they could do better without. Ben Franklin read the original in French in 1759 and persuaded the publisher to prepare a hurried English translation. Franklin sent three copies of the translation to his colleagues in the Colonies in the early 1760s, including the Adams cousins and Jefferson. Jefferson made Vattel our first law text in 1779 at our first law school, at William and Mary, whose curriculum he designed.
The only reason our framers included natural-born citizens, a class understood for thousands of years, was to require that our president, and only our president and vice president, be natural born. Thus for purposes of the Constitution, you were natural-born, or an alien, probably naturalized by the laws of the state in which you lived or into which you immigrated, every state having differences in their naturalization laws. If you were born to citizens of that state, which couldn't have happened before 1776 (before that everyone was a subject, critically different from a citizen) on our soil, you were natural-born.
What about the early presidents? That was the “grandfather clause” of Article II. The 14 year residency requirement implied that if you were 35 years old when the Constitution was signed in 1787, you were 21 years old in 1773, eligible to be drafted, and have lived through the terror of the revolutionary war. If you didn't fight, you supported the effort because Royalists were vilified, many killed, and returned to England rather than stay. The founders fought for their freedom. Those naturally born were assumed to be the children of those who fought (See Dr. David Ramsay's “Dissertation on Citizenship”)
Upon reflection, our astute framers may have thought natural-born citizenship a strategic defense, since the founders knew why they fought, and risked their own lives, their families, and everything they worked for to win freedom. What might they do to try to insure the allegiance of the person with the most power in the nation? When we see our supposed defenders of constitutional originalism floating one naturalized citizen after another, Jindal, Rubio, Cruz, there are clearly forces, probably avarice for politicians, working to weaken our foundation. We have many more natural born citizens than naturalized citizens. Suzanna Martinez is natural born, Sarah Palin is natural born, Paul Ryan is natural born, ...