Flight director was a title held by a Starfleet officer at Starfleet Command's Fleet Operations Center, Sol Sector.
In January of 2155, the flight directors were L. Burton, M. Howard, P. Lauritson, and M. Rush. This title and the officer names were listed on the directory placards. (ENT: "Demons", directories)
Flight Director was a reference to NASA and ISS terminology.
Burton and Rush were listed on the Starfleet directory and Howard and Lauritson were listed on the United Earth directory.