Of course, and anyone else succeeding to the office of President. (That is if the Speaker of the House would be required to succeed to the office, she would need to be a Natural born citizen, else the next in line would do so in her stead.)
You are making the assumption, without much evidence that I'm aware of, that there is a difference.
From the majority opinion inWong Kim Ark:
Mr. Binney in the second edition of a paper on the Alienigenae of the United States, printed in pamphlet at Philadelphia, with a preface bearing his signature and the date of December 1, 1853, said: 'The common- law principle of allegiance was the law of all the states at the time of the Revolution and at the adoption of the constitution; and by that principle the citizens of the United States are, with the exceptions before mentioned [namely, foreign-born children of citizens, under statutes to be presently referred to], such only as are either born or made so, born within the limits and under the jurisdiction of the United States, or naturalized by the authority of law, either in one of the states before the constitution, or, since that time, by virtue of an act of the congress of the United States.' Page 20. 'The right of citizenship never descends in the legal sense, either by the common law, or under the common naturalization acts. It is incident to birth in the country, or it is given personally by statute. The child of an alien, if born in the country, is as much a citizen as the natural-born child of a citizen , and by operation of the same principle.' [169 U.S. 649, 666] Page 22, note.
This certainly implies that the "child of an alien, if born in the country" and "natural born child of a citizen" are not the same thing, and in fact gives a basic definition of "Natural Born". That is "child of a citizen'"
I believe you are misreading the sentence. Notice he does not say "natural-born citizen," but "natural-born child of a citizen."
More critically he says the child of a citizen and the child of an alien are both citizens, by operation of the same principle.
IOW, they are citizens for the same reason and of the same type. There is no difference between them.
Native born = natural born, whether the parents are citizens or aliens.
The common-law principle of allegiance was the law of all the states at the time of the Revolution and at the adoption of the constitution
As I hope you know, the English common-law principle of allegiance and citizenship is that citizenship springs from location of birth, not ancestry. This is in direct contrast to de Vattel and his continental European civil "Law of Nations" in which citizenship is inherited from the parents.
Some on FR keep trying to drag de Vattel into the issue, when Blackstone is a far better authority on what the Founders would have taken natural-born to mean.