(b) An agreement that authorizes the county treasurer to collect taxes jointly and concurrently with the tax collecting officer of such city, town, village or school district shall have no effect upon the tenure, powers or duties of the incumbent tax collecting officer, except that the county treasurer shall also be considered a tax collecting officer of the city, town, village or school district, with all the powers and duties thereof. In no case shall such an agreement be construed to empower the tax collecting officer of a city, town, village or school district to collect taxes that have been returned to the county treasurer as unpaid.

(c) An agreement that delegates to the county treasurer the sole and exclusive authority to collect taxes for such city, town, village or school district shall have the effect of making the county treasurer the sole tax collecting officer of such city, town, village or school district, and of abolishing the separate office of tax collecting officer in such city, town, village or school district, for as long as the agreement shall be in effect. Provided, however, that if the office to be abolished is elective, the agreement shall be submitted for the approval of the electors of the contracting city, town or village in the manner provided by section twenty-three of the municipal home rule law. Provided further, that such an agreement shall not take effect during the term of an incumbent tax collecting officer, unless the office should become vacant prior to the expiration of such term. Upon the termination of such an agreement, the office of tax collecting officer shall be deemed reestablished as an appointive office of the city, town, village or school district, unless such office has been abolished or its functions transferred to another officer pursuant to law.