Correct as to the diplomat; incorrect as to the illegal immigrant mother.
The law on this subject is clear. You may say that the Constitution doesn’t support this view, but in fact it does. “Subject to the jurisdiction” of a court of law means that the court may issue and enforce orders against a person. The diplomat is free from such jurisdiction; the illegal immigrant is not. If the dilpomat were to go on a killing spree, she could be expelled from the country, but not tried for murder; if the illegal immigrant went on a killing spree, she could be arrested, prosecuted, and jailed.
If a person is required to obey the orders of a court, they are under the jurisdiction of that court.
And the Supreme Court has answered these questions directly. In US v. Wong Kim Ark, 169 U.S. 649 (1898), as Really Not All That Bright mentioned above, the Court held that the Fourteenth Amendment granted citizenship to the child of legal resident aliens. They didn’t go quite as far as RNATB suggested, since their holding applied to legal residents. But that’s not the end of the discussion: In Plyler v. Doe, 457 U.S. 202 (1982), the Court addressed the jurisdiction question head on:
Finally, in INS vs. Rios-Piñeda, 471 U.S. 444 (1985), the Court answered your precise claim: a child born in this country to illegal immigrants is a citizen.