These two types of citizen are not the same, and a native-born citizen is not eligible to be president.
“I beg to differ. A natural-born citizen is one who was born on United States soil to two American citizens. “
You are wrong.
The Wong Ark ruling in multiple places used the phrase natural-born as a phrase to clearly mean citizen (or subject) from time of birth, and cited laws where children of aliens born in the realm were natural-born citizens/subjects.
Ill share an extended quote from the Supreme Court Wong Kim Ark ruling to show what I mean. The distinction you claim between natural-born and some other class of citizen at birth is found nowhere in statutes or Supreme Court rulings, and is certainly at variance with the intent of the 14th amendment.
http://en.wikisource.org/wiki/United_States_v._Wong_Kim_Ark/Opinion_of_the_Court
The English statute of 11 & 12 Will. III (1700). c. 6, entitled
An act to enable His Majestys natural-born subjects to inherit the estate of their ancestors, either lineal or collateral, notwithstanding their father or mother were aliens,
enacted that all and every person or persons, being the Kings natural-born subject or subjects, within any of the Kings realms or dominions, might and should thereafter lawfully inherit and make their titles by descent to any lands
from any of their ancestors, lineal or collateral, although the father and mother, or father or mother, or other ancestor, of such person or persons, by, from, through or under whom
title should be made or derived, had been or should be born out of the Kings allegiance, and out of is Majestys realms and dominions, as fully and effectually, as if such parents or ancestors had been naturalized or natural-born subject or subjects within the Kings dominions. 7 Statutes of the Realm, 90. It may be observed that, throughout that statute, persons born within the realm, although children of alien parents, were called natural-born subjects. As that statute included persons born within any of the Kings realms or dominions, it, of course, extended to the Colonies, and, not having been repealed in Maryland, was in force there. In McCreery v. Somerville, (1824) 9 Wheat. 354, which concerned the title to land in the State of Maryland, it was assumed that children born in that State of an alien who was still living, and who had not been naturalized, were native-born citizens of the [p662] United States, and, without such assumption, the case would not have presented the question decided by the court, which, as stated by Mr. Justice Story in delivering the opinion, was
whether the statute applies to the case of a living alien ancestor, so as to create a title by heirship where none would exist by the common law if the ancestor were a natural-born subject.
9 Wheat. 356.
Again, in Levy v. McCartee (1832), 6 Pet. 102, 112, 113, 115, which concerned a descent cast since the American Revolution, in the State of New York, where the statute of 11 & 12 Will. III had been repealed, this court, speaking by Mr. Justice Story, held that the case must rest for its decision exclusively upon the principles of the common law, and treated it as unquestionable that, by that law, a child born in England of alien parents was a natural-born subject, quoting the statement of Lord Coke in Co.Lit. 8a, that,
if an alien cometh into England and hath issue two sons, these two sons are indigenae, subjects born, because they are born within the realm,
and saying that such a child was a native-born subject, according to the principles of the common law stated by this court in McCreery v. Somervlle, 9 Wheat. 354.