(a) The Civil Service. The civil service of the City includes all employees of the City, full or part-time, who are paid any compensation out of public funds for the performance of their services. The civil service of the City is divided into two classes:
(1) The classified civil service, who are employees of the City to whom all of the rights, privileges and duties of the civil service laws, rules and regulations of the City appertain, and
(2) The unclassified civil service, who are employees of the City not amenable to any of the provisions of the civil service laws, rules or regulations of the City.
(b) Classified Service.
(2) By collective bargaining agreement. All employees of the City who are members of a recognized bargaining unit, and who are beneficiaries of a collective bargaining agreement with the City, which agreement invokes and confers upon its beneficiaries the protection of the Civil Service Commission of the City, are members of the classified service.
(3) By action of the Commission. Except those positions specifically exempted by law, such employees of the City as the Commission may find, upon investigation and specific determination, to be in a class or group which should be amenable to the protection of the Civil Service Commission of the City are members of the classified service, provided however, that such finding and determination by the Commission be approved and ratified by the Council, two-thirds of all its members agreeing thereto.
(c) Unclassified Service. All employees of the City, including those persons holding positions specifically exempted at law from the application of the civil service rules of the City, who are not specifically identified in Section 135.05
(b)(1), (2), or (3) above, are members of the unclassified service of the City.