If we include both President Lincoln and President Hayes, there were only four presidents in this group. Of these four, I would argue that President Lincoln did the best job of fulfilling and respecting the president’s job as outlined in Article II of the Constitution. This is, at least in part, because he was president during a time of crisis and war.
I base this answer largely on what is in Section 2, Clause 1 of Article II. There, we are told that the president is supposed to act as the Commander in Chief of the armed forces of the United States. As president during the Civil War, President Lincoln did a very good job of this. Though he was a civilian with little military background, he did not let the military dictate policy to him. Instead, he was firmly in control of the military, pushing it to do what he believed was necessary to win the war.
In this same clause, we are also told that the president can require cabinet members to give him (or someday her) their opinions on various matters. This at least implies that the president should consider the opinions of the people in the cabinet, but should be in ultimate charge of making the decisions. Lincoln is known to have done this very well. He assembled a “team of rivals” to make sure that he would get honest and varying opinions. He did not surround himself with “yes men.” By doing this, he was, in my view, acting in the way a president should.
Thus, Lincoln very clearly respected the office by making sure that he did not give up power to the military and by acting as an arbitrator for the cabinet members that he had assembled.