Since this group of presidents includes President Nixon, it is easy to choose which president did a poor job of respecting the job of the president as outlined in Article II of the Constitution. That article says that one of the president’s jobs is to see that the laws of the country are faithfully executed. This presumably means that presidents should obey the laws and should try their best to make sure that the laws are carried out. President Nixon clearly did a very poor job of this. We can say this because he used his position to try to obstruct justice. That is, he tried to use his powers as president to make sure that the laws could not be enforced against the people who were involved in the Watergate break-in. More than perhaps any other president, he directly and intentionally broke his promise to carry out the laws of the United States.
This leaves us three presidents to pick from for the question of which president did a good job. I do not think that any of these presidents was notably better than the others. However, I suppose you could argue that President Johnson did a good job because of his actions as commander-in-chief. In our Constitution, the president is made commander-in-chief so that the civilian government will be in charge of military operations. We do not want an independent military that makes foreign policy decisions on its own. Instead, we want our elected officials to be in charge. Johnson did a good job (at least arguably) in making sure that he controlled the war in Vietnam. For example, he did not allow the military to decide how aggressively the US would fight. Instead, he kept careful control over the war effort to ensure that the US would fight only a limited war. This hurt our ability to win the war, but it did preserve civilian control of the military and it kept the war from spreading to involve countries like China. For this reason, you can at least argue that Johnson did a good job of respecting the office.
This is a pretty interesting but rather tricky question to answer. For one, there are only four presidents to choose from, and two of them didn't even serve an entire term. Also, you're asking for which one best respected the defined job of the presidency, not necessarily which one was the best, or which one executed the role most effectively. It also doesn't help that the duties of the president are very broadly stated and very few in the Constitution.
However, I would say that John F. Kennedy did the best job of executing the role of the president as stated in Article II of the Constitution. For one, the president is given the power to make treaties, and Kennedy did this at a rate quicker than any of the other presidents. In his less than three years in office, he signed ten treaties, including the Arms Control and Disarmament Agency and the Vienna Convention on Diplomatic Relations. As Commander in Chief, he used the military in ways he deemed best. This was shown in his deployment of military advisers in Vietnam, and his use of a naval quarantine in handling the Cuban Missile Crisis.
The president who I feel did the worst job of respecting the office is Richard Nixon. He of course was part of the Watergate scandal, and I feel that did not "take Care that the Laws be faithfully executed." The president is also assigned to make appointments to the Supreme Court and this is something Nixon struggled with as well. He had a hard time getting his appointments passed, such Harrold Carswell, who was a racist. Though this isn't directly stated, it is inferred that the president needs to be able to work with Congress, which Nixon did so very poorly. Nixon, who was a Republican, had to deal with a Democratic Congress, and couldn't get potentially major acts, like the Family Assistance Program, through.