Lets look a little closer.
Justice Gray quotes Binney who makes a comparison between a jus soli citizen versus a natural born citizen.
"Mr. Binney in his essay before quoted, "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." It can hardly be denied that an alien is completely subject to the political jurisdiction of the country in which he resides -- seeing that, as said by Mr. Webster, when Secretary of State, in his Report to the President on Thrasher's Case in 1851, and since repeated by this court,"
We see Justice Gray is comparing a "citizen," a jus soli citizen, to a natural born citizen. Binney or Gray would not consider Obama as a natural born citizen.
Moving on what Gray said in WKA,
"The child born of alien parents in the United States is held to be a citizen thereof, and to be subject to duties with regard to this country which do not attach to the father.
The same principle on which such children are held by us to be citizens of the United States, and to be subject to duties to this country, applies to the children of American fathers born without the jurisdiction of the United States, and entitles the country within whose jurisdiction they are born to claim them as citizen and to subject them to duties to it."
Justice Gray never comes close to saying that children born of alien parents are natural born citizens. Far from it. It further stated that the same principle applies to children born to American parents in foreign countries. The foreign state can claim them as citizens. There is no doubt that Wong Kim Ark was not a natural born citizen, and neither is Barack Obama based on Gray's Supreme Court opinion.
From the 1771 Edition Encyclopedia Britannica. You'll see that a "NATIVE, a person considered as born in a certain place which was the proper residence of his parents." In other words, he is born in the country with citizen parents.
The proper residence of Wong Kim Arks PARENTS was China. The proper resident of Barack Obama's FATHER was Kenya. Obama in 1771 would not have been considered a native born, or would Obama have been called a natural born citizen.
Fast forward to 1952 in the Supreme Court case Kawikita v. United States:
"U.S. Supreme Court
Kawakita v. United States, 343 U.S. 717 (1952)
Kawakita v. United States
At petitioner's trial for treason, it appeared that originally he was a native-born citizen of the United States and also a national of Japan by reason of Japanese parentage While a minor, he took the oath of allegiance to the United States; went to Japan for a visit on an American passport, ..."
And
"First. The important question that lies at the threshold of the case relates to expatriation. Petitioner was born in this country in 1921 of Japanese parents who were citizens of Japan., He was thus a citizen of the United States by birth, Amendment XIV, § 1 and, by reason of Japanese law, a national of Japan. See Hirabayashi v. United States, 320 U. S. 81, 320 U. S. 97.and law. While a minor, he took the oath of allegiance to the United States; went to Japan for a visit on an American passport, and was prevented by the outbreak of war from returning to this country. During the war, he reached his majority in Japan, changed his registration from American to Japanese, showed sympathy with Japan and hostility to the United States,..."
Here we see SCOTUS in the modern usage of "native-born." The Supreme Court further stated that Kawakita was a citizen by virtue of the 14th Amendment, and I have no doubt because of the 1898 Wong Kim Ark ruling. Kawakita had duel allegiances. Would a natural born citizen have to take the oath of allegiance to the United States as Kawakita did?