Is a person's admission of quitting masturbation, after the questioner deduced that he practiced it, considered a form of publicizing a sin, and what would be the answer if the question were repeated?

It is reprehensible for a Muslim to inform others of his sins. The basic principle is for a Muslim to conceal his own faults, unless the disclosure is for a valid need and recognized benefit, such as seeking a fatwa (definition). In such a case, there is no harm. If a Muslim is asked about a sin he has repented from, and he believes the questioner has knowledge of it, and he informs him of his cessation from it, this is not considered forbidden public disclosure, and it is hoped that there will be no blame upon him for doing so.