Except as otherwise provided in the case of an elector whose name has not been placed on or has been removed from the enrollment list under section 9-59, 9-60, 9-61 or 9-62, any elector not enrolled on any enrollment list may at any time make a written and signed application for enrollment to the registrars of voters on an application form for admission as an elector, in accordance with the requirements of this section. The application shall be effective as of the date it is filed with the registrars of voters of the town of residence of the applicant and any person making application for enrollment in such manner shall immediately be entitled to the privileges of party enrollment unless the application for enrollment (1) is filed in person by the applicant with the registrars of voters after twelve o’clock noon on the last business day before a primary, in which case he shall be entitled to the privileges of party enrollment immediately after the primary, (2) is otherwise filed with the registrar after the fifth day before the primary, in which case he shall be entitled to the privileges of party enrollment immediately after the primary, except as provided in section 9-23a, or (3) is filed with the registrars of voters after 5:00 p.m. on the last business day before a caucus or convention, in which case he shall be entitled to the privileges of party enrollment immediately after the caucus or convention. The application shall be signed or initialed by the registrar, deputy, assistant or registrar’s clerk receiving it, or by such other personnel as such registrar or deputy may appoint for the purpose, showing the date when such application is received and, in the case of an applicant not immediately eligible under section 9-59, 9-60, 9-61 or 9-62 to the privileges accompanying enrollment in the party named in his application, the date upon which such applicant becomes so eligible. In municipalities divided into voting districts in which an enrollment session is held in each district thereof under section 9-51, application for enrollment shall be made to the registrar or assistant registrar, as the case may be, in the voting district in which such elector is entitled to vote at the time of making such application. If any registrar or assistant registrar fails to add any name to any such list on written application or adds any name to any such list except as herein provided, he shall be guilty of a class D misdemeanor.