No. It states that but then proceeds to say that there are questions as to whether one born in the US but of non-citizens is natural born. It ultimately doesn’t resolve that issue but only says there is less support for it. It merely states that it is unquestionable that US born children of US citizens are natural born (they also qualify as native born) but some have questioned whether one who is born in the US but not of citizens is natural born. It doesn’t say one way or the other on the issue just that some have argued over it and thus is it is less established or clear than if both requirements were met. And again, this is just dicta anyway.
No it doesn't. It says there are doubts about the CITIZENSHIP, not whether they are natural born. Read it:
Some authorities go further and include as citizens children born within the jurisdiction without reference to the citizenship of their parents. As to this class there have been doubts, but never as to the first.
Note it says includes AS CITIZENS. It does NOT says includes as "natural born citizens." The previous definition was also framed as just becoming citizens:
At common-law, with the nomenclature of which the framers of the Constitution were familiar, it was never doubted that all children born in a country of parents who were its citizens became themselves, upon their birth, citizens also.
BOTH definitions are about only about persons becoming citizens, not specifically about becoming natural-born citizens. The NBC part was added as a characterization because that definition matches the law of nations definition that says this same thing, and because there is no doubt about this class of persons.