Justice Gray quotes English common law not as support for the ruling in Wong Kim Ark but rather to compare and contrast the various historical definitions internationally of "natural-born." He further refers to the historical concept of a person's political status versus civil status noting that the two are distinctly separate by definition but can overlap in practice.
Your question "Which one is correct? And why?" essentially makes my point that a SCOTUS ruling is needed because it isn't clear whether or not a person born on U.S. soil to a foreign citizen parent is a natural-born citizen. Ark affirms only that such a person is a citizen.
So please point out to me what section of the Constitution identifies three classes?
By using three different labels for citizenship (natural-born citizen, citizen, naturalized citizen) the language of the Constitution inherently identifies three entities, granted those entities may overlap in some cases. If there were no difference between a citizen and a natural-born citizen, why distinguish a different standard for President?
Here's an analogy on my take regarding a citizen versus a natural-born citizen. All managers are employees of a company, but not all employees are managers. There's a higher requirement for management and greater assigned responsibility. Both are entitled to the benefits conferred upon employees by the company and both are bound by the rules and policies of the company. The designation of employee is equivalent to one's civil status - being bound by a set of rules and eligible to the benefits of employment. The designation of manager is equivalent to one's political status - owing a greater responsibility to the company's business objectives and goals.
But until then federal law says that children born in the U.S. are natural born citizens regardless of their parent's nationality.
Nowhere does federal law stipulate that anyone is a natural-born citizen. The courts have refused to define "natural-born citizen" and have gone only so far as to clarify who is and is not a citizen.
So, in short, perhaps the label “natural-born” citizen was intended to be more of a political distinction than it was a civil distinction. There really is no civil distinction between a citizen and a naturalized citizen as both are entitled to the same protection under the law. There is however a political distinction between the two.