Who determines when the flags are lowered to half-staff, and for

According to the American Legion website, "On Memorial Day the flag should be displayed at half-staff until noon only, then raised to the top of the staff. By order of the President, the flag shall be flown at half-staff upon the death of principal figures of the United States Government and the Governor of a State, territory, or possession, as a mark of respect to their memory. In the event of the death of other officials or foreign dignitaries, the flag is to be displayed at half-staff according to Presidential instructions or orders, or in accordance with recognized customs or practices not inconsistent with law. In the event of the death of a present or former official of the government of any State, territory, or possession of the United States or the death of a member of the Armed Forces from any State, territory, or possession who dies while serving on active duty, the Governor of that State, territory, or possession may proclaim that the National flag shall be flown at half-staff and the same authority is provided to the Mayor of the District of Columbia with respect to present or former officials of the District of Columbia and members of the Armed Forces from the District of Columbia. When the Governor of a State, territory, or possession, or the Mayor of the District of Columbia, issues a proclamation under the preceding sentence that the National flag be flown at half-staff in that State, territory, or possession or in the District of Columbia because of the death of a member of the Armed Forces, the National flag flown at any Federal installation or facility in the area covered by that proclamation shall be flown at half-staff consistent with that proclamation. The flag shall be flown at half-staff 30 days from the death of the President or a former President; 10 days from the day of death of the Vice President, the Chief Justice or a retired Chief Justice of the United States, or the Speaker of the House of Representatives; from the day of death until interment of an Associate Justice of the Supreme Court, a Secretary of an executive or military department, a former Vice President, or the Governor of a State, territory, or possession; and on the day of death and the following day for a Member of Congress. The flag shall be flown at half-staff on Peace Officers Memorial Day, unless that day is also Armed Forces Day."