Some USCIS forms are called "petitions" (e.g. I-130, I-140, I-129), while others are called "applications" (e.g. N-600, N-400, I-485). Petitions have "petitioners", applications have "applicants". N-600 is an "application", so it has an applicant, not a petitioner.
My understanding is that the person for whom a certificate of citizenship is requested is considered the applicant for N-600, even if that person is a minor and the form is submitted by a parent on the child's behalf. E.g. I'd expect that your son's name will be listed as the "applicant" on the N-600 receipt notice, and your name will be listed as "c/o".
But it may well be that the distinction is academic in this case and hopefully they'll send you an e-mail notification anyway.
Legally speaking the person seeking the certificate via the N-600 is seeking
recognition of citizenship already attained by law and is therefore a "claimant" because they are making a citizenship claim. In INA legal terms,
all citizens are nationals but not all nationals are citizens, the legal right to assert one's specific claim as applicable is addressed in the single statutory concept of
nationality.
This concept is most clear in the petitions for review of orders of removal when one seeks to avoid deportation by asserting a
claim to nationality before the Circuit Court of Appeals:
INA 242 JUDICIAL REVIEW OF ORDERS OF REMOVAL
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(b)
Requirements for Review of Orders of Removal.-With respect to review of an order of removal under subsection (a)(1), the following requirements apply:
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(5)
Treatment of nationality claims.-
(A)
Court determination if no issue of fact.-If the petitioner
claims to be a national of the United States and the court of appeals finds from the pleadings and affidavits that no genuine issue of material fact about the petitioner's nationality is presented,
the court shall decide the nationality claim.
(B)
Transfer if issue of fact.-If the petitioner
claims to be a national of the United States and the court of appeals finds that a genuine issue of material fact about the petitioner's nationality is presented, the court shall transfer the proceeding to the district court of the United States for the judicial district in which the petitioner resides for a new
hearing on the nationality claim and a decision on that claim as if an action had been brought in the district court under section 2201 of title 28, United States Code.
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When an N-600 is denied it has a specific administrative appeal to AAO on form I-290B and then a further Judicial Appeal path (in that respect it is similar to N-400 applicants who also have specific statutory appeals rights starting with the N-336).
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INA 360 [8 U.S.C. 1503] JUDICIAL PROCEEDINGS FOR DECLARATION OF UNITED STATES NATIONALITY IN EVENT OF DENIAL OF RIGHTS AND PRIVILEGES AS NATIONAL
(a) If any person who is within the United States
claims a right or privilege as a national of the United States and is denied such right or privilege by any department or independent agency, or official thereof, upon the ground that he is not a national of the United States, such person may institute an action under the provisions of section 2201 of title 28, United States Code, against the head of such department or independent agency for a judgment declaring him to be a national of the United States, except that no such action may be instituted in any case if the issue of such person's status as a national of the United States (1) arose by reason of or in connection with any removal proceeding under the provisions of this or any other act, or (2) is in issue in any such removal proceeding. An action under this subsection may be instituted only within five years after the final administrative denial of such right or privilege and shall be filed in the district court of the United States for the district i n which such person resides or claims a residence, and jurisdiction over such officials in such cases is hereby conferred upon those courts.
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However, when the claimant is a minor (or incompetent) someone will be acting on their behalf. So, it makes no difference who is listed on the G-1145 which is merely a uniform simple form to allow for the collection of an e-mail address. As forms are updated they will eventually all include a space for that e-mail address and even further down the road, e-filing will take over.