Library districts are governed by a five member Board of Directors, which are elected.
Directors are elected for five-year terms, with one term expiring every year.
The Board of Directors is completely independent of any city government.
Funding for the district comes directly to the district from the County Treasurer, and is completely independent of any city government.
(Note: After the November election, the very first Board of Directors is appointed by the Payette County Commissioners - because the next election won't occur until May, at which point the term-cycle will actually start.)
City-funded libraries are managed by a Board of Trustees.
Board members are appointed by the mayor (there are no elections).
While the Board of Trustees makes the policy decisions of the library, it is still at the mercy of the City Council for funding. At any time, the City Council could legally stop funding their library (it might be a bad PR move, but it's well within their right to do so - as the library is considered like any other city department).