When the President takes the oath of Office, the oath is binding. "I do solemnly swear (or affirm) that I will faithfully execute the Office of President of the United States, and will to the best of my Ability, preserve, protect and defend the Constitution of the United States." (United States Constitution, Article II, Section 1.)
When the President repeats the Oath of Office and is sworn in as President, the requirement is to tell the truth in all matters, and is held accountable in all statements, with the possible exception of national security matters. The President's responsibility is to "preserve, protect and defend the Constitution of the United States." The President of the United States is required as a "High Official" to speak the truth in all matters and is held accountable to do so throughout the length of that individual's term in the Office of President.
The use of the word "high crimes" does not necessarily mean a crime that is "more serious" than other crimes. It refers to those punishable offenses that can only be applied to persons in the highest of offices. In the case of the President, because of the official status under which that person holds the office, the standard is meant for that person who holds special obligations that the ordinary person is not under. Examples might be the action, or inaction, taken by the person in such high office.
Some might argue if the President failed to warn people of the danger that existed when he knew of a terrorist attack that would take the lives of American citizens, it could be construed that he did not act in the best interest of the citizens and put them in danger and thereby failed to protect the citizens from the harm he knew would cause their death. An ordinary citizen with no official responsibilities would not be under the same standard of "high crimes."
Some might argue that specific actions taken by the President might be seen as "high crimes" if those actions involved the President's actual knowledge of the dangers of the action and the likely consequences or outcomes of the action relative to the action being a direct threat to the welfare of the citizens of the United States. For example, if the President had prior knowledge of how a subversive group would use various weapons or instruments of war, and that use constituted a direct attack on the sovereignty of the United States with the likely outcome being the death of American citizens, following through with providing such weapons or instruments of war to that subversive group could be construed as committing "high crimes."
Under such circumstances the United States Congress through the House of Representatives are authorized under the Constitution to initiate impeachment proceedings. "The President, Vice President, and all civil Officers of the United States shall be removed from Office on Impeachment for, and Conviction of, Treason, Bribery, or other High Crimes and Misdemeanors." (United States Constitution, Article II, Section 4.)
Americans need to remain vigilant to their watch of the highest officials in the federal government to ensure they abide by the oath or affirmation they have taken and hold them accountable when they fail in their responsibilities, this responsibility includes the President.