There were originally two Sashes of Rassilon, and were first intended to be personal force shields. Rassilon gave one to Omega and kept the other for himself. (PROSE: The Scrolls of Rassilon)
It gave those who wore it protection from the gravitational and energy forces which emanated from the Eye of Harmony. It also offered some control over the Eye itself. However, it had less power than its reputation suggested and would not have saved the Master had he opened the Eye fully. (TV: The Deadly Assassin)
The Presidents are categorised by in-universe chronological order. Loose chronological order applies for the lists, but when in doubt (as with the temporal uncertainty of the Time Wars), list them in the order in which they were first depicted as Presidents in the real world.