The services of a Hudson River pilot and a Long Island Sound-Block Island Sound pilot in arriving vessels end and a pilot is entitled to discharge when the vessel has been brought to a safe anchorage or the vessel is securely moored at its berth. Hudson River pilots and Long Island Sound-Block Island Sound pilots shall supervise the mooring; the departing from a mooring berth; and shifting from one berth to another except when a vessel is warped from one berth along a wharf to another using mooring lines and deck machinery. When requested by the master to remain with a vessel at an anchorage or berth, the Hudson River pilot or Long Island Sound-Block Island Sound pilot shall do so or arrange for a relief and shall not leave until the relief arrives. A pilot shall not leave a vessel underway unless the pilot has been properly relieved.