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Are you saying that someone who is a “citizen at birth” is NOT a natural-born citizen?

The following state (from Title 8 of the U.S. code) lists categories of people who are “United States citizens at birth.”

Subsection (g) is relevant here. (sorry for the length)

Page 391 TITLE 8—ALIENS AND NATIONALITY § 1401
nonimmigrants described in section
1101(a)(15)(L) of this title, including the number
of such nonimmigrants who are classified on the
basis of specialized knowledge and the number
of nonimmigrants who are classified on the basis
of specialized knowledge in order to work primarily
at offsite locations.
(b) Applicability
Subsection (a) of this section shall apply to
petitions filed on or after the effective date of
this subtitle.
(Pub. L. 108–447, div. J, title IV, § 414, Dec. 8, 2004,
118 Stat. 3352.)
REFERENCES IN TEXT
This subtitle, referred to in subsec. (b), means subtitle
A (§§ 411–417) of title IV of div. J of Pub. L. 108–447.
For the effective date of subtitle A, see section 417 of
Pub. L. 108–447, set out as an Effective Date of 2004
Amendment note under section 1184 of this title.
CODIFICATION
Section was enacted as part of the L–1 Visa (Intracompany
Transferee) Reform Act of 2004, and also as
part of the L–1 Visa and H–1B Visa Reform Act and the
Consolidated Appropriations Act, 2005, and not as part
of the Immigration and Nationality Act which comprises
this chapter.
EFFECTIVE DATE
Section effective 180 days after Dec. 8, 2004, see section
417 of Pub. L. 108–447, set out as an Effective Date
of 2004 Amendment note under section 1184 of this title.
§ 1381. Secretary of Labor report
Not later than January 31 of each year, the
Secretary of Labor shall report to the Committees
on the Judiciary of the Senate and the
House of Representatives on the investigations
undertaken based on—
(1) the authorities described in clauses (i)
and (ii) of section 1182(n)(2)(G) of this title;
and
(2) the expenditures by the Secretary of
Labor described in section 1356(v)(2)(D) of this
title.
(Pub. L. 108–447, div. J, title IV, § 424(c), Dec. 8,
2004, 118 Stat. 3356.)
CODIFICATION
Section was enacted as part of the H–1B Visa Reform
Act of 2004, and also as part of the L–1 Visa and H–1B
Visa Reform Act and the Consolidated Appropriations
Act, 2005, and not as part of the Immigration and Nationality
Act which comprises this chapter.
EFFECTIVE DATE
Section effective 90 days after Dec. 8, 2004, see section
430 of Pub. L. 108–447, set out as an Effective Date of
2004 Amendment note under section 1182 of this title.

SUBCHAPTER III—NATIONALITY AND
NATURALIZATION
PART I—NATIONALITY AT BIRTH AND COLLECTIVE
NATURALIZATION
§ 1401. Nationals and citizens of United States at
birth
The following shall be nationals and citizens
of the United States at birth:
(a) a person born in the United States, and
subject to the jurisdiction thereof;
(b) a person born in the United States to a
member of an Indian, Eskimo, Aleutian, or
other aboriginal tribe: Provided, That the
granting of citizenship under this subsection
shall not in any manner impair or otherwise
affect the right of such person to tribal or
other property;
(c) a person born outside of the United
States and its outlying possessions of parents
both of whom are citizens of the United States
and one of whom has had a residence in the
United States or one of its outlying possessions,
prior to the birth of such person;
(d) a person born outside of the United
States and its outlying possessions of parents
one of whom is a citizen of the United States
who has been physically present in the United
States or one of its outlying possessions for a
continuous period of one year prior to the
birth of such person, and the other of whom is
a national, but not a citizen of the United
States;
(e) a person born in an outlying possession of
the United States of parents one of whom is a
citizen of the United States who has been
physically present in the United States or one
of its outlying possessions for a continuous period
of one year at any time prior to the birth
of such person;
(f) a person of unknown parentage found in
the United States while under the age of five
years, until shown, prior to his attaining the
age of twenty-one years, not to have been born
in the United States;
(g) a person born outside the geographical
limits of the United States and its outlying
possessions of parents one of whom is an alien,
and the other a citizen of the United States
who, prior to the birth of such person, was
physically present in the United States or its
outlying possessions for a period or periods totaling
not less than five years, at least two of
which were after attaining the age of fourteen
years: Provided, That any periods of honorable
service in the Armed Forces of the United
States, or periods of employment with the
United States Government or with an international
organization as that term is defined
in section 288 of title 22 by such citizen parent,
or any periods during which such citizen parent
is physically present abroad as the dependent
unmarried son or daughter and a member
of the household of a person (A) honorably
serving with the Armed Forces of the United
States, or (B) employed by the United States
Government or an international organization
as defined in section 288 of title 22, may be included
in order to satisfy the physical-presence
requirement of this paragraph. This proviso
shall be applicable to persons born on or
after December 24, 1952, to the same extent as
if it had become effective in its present form
on that date; and
(h) a person born before noon (Eastern
Standard Time) May 24, 1934, outside the limits
and jurisdiction of the United States of an
alien father and a mother who is a citizen of
the United States who, prior to the birth of
such person, had resided in the United States.
(June 27, 1952, ch. 477, title III, ch. 1, § 301, 66
Stat. 235; Pub. L. 89–770, Nov. 6, 1966, 80 Stat.