I think that Eisenhower and Reagan probably would have been tempted to seek a third term, if possible. They both had health problems during their Presidencies, but I could see Eisenhower seeking a third term anyway. He had a difficult time stepping away, which is one reason why he waited so long to give Richard Nixon a solid endorsement in 1960. It wasn’t necessarily a lack of confidence in Nixon’s abilities, but partly because Ike felt that he (Ike) was still the best man for the job.
Reagan, like Clinton, loved being President, too. But when Reagan left office in 1989, he was about two weeks away from his 78th birthday and, according to his official biographer, Edmund Morris, there were signs that he may have been facing the early stages of his Alzheimer’s in the last few weeks of his Administration. Since President Reagan looked relatively healthy and definitely looked fit for his age, it’s difficult for people to realize that he was almost a full eight years older than Eisenhower (70) was when Ike left office. Even if Eisenhower had served another term, Ike still would have been four years younger than Reagan at the end of that third term. I think Reagan’s age and deteriorating health would have prevented him from a third term if it was Constitutionally possible. As closely as his public image was protected by Nancy Reagan, there is no way she would have stood by while he hung on for another term and publicly started to suffer from serious Alzheimer’s symptoms.
An interesting thing is that, if they had the opportunity to run for a third term and their health allowed it, I think Eisenhower, Reagan, and Clinton all would have been easily elected to another term. I think George W. Bush would have had a much more difficult time with seeking a third term, if possible. However, I don’t think Bush would have run again even if he was Constitutionally eligible. In those last few months of 2008, President Bush looked SO ready to get back to Texas. Even if his chances of being re-elected were positive, I still think he would have chosen retirement instead of a third term.
As for the second part of your question, I think that Truman would have stepped away in 1952, no matter what. All Truman ever wanted to do was remain a U.S. Senator. When he was suggested as a potential Vice Presidential candidate, he was not interested, and when others reminded him that President Franklin D. Roosevelt likely wouldn’t survive the term, Truman declared that he didn’t want to be President either. Of course, he was elected Vice President and as in the case of almost every VP who succeeds to the Presidency, once Truman got to the White House he wanted to be elected to a term in his own right. Still, before Eisenhower declared that he was a Republican, Truman was suggesting that he (Truman) would be happy to step aside and be Eisenhower’s running mate if Ike wanted to run for President as a Democrat. So, Harry Truman did not mind retiring home to Missouri in 1952, and I think he would have done so, no matter what.
LBJ’s case was different. The fact that he was very nearly upset in the 1968 New Hampshire Democratic Primary by Eugene McCarthy really shook President Johnson up and showed that he was vulnerable. If there wasn’t a serious challenge from within his own party — first from McCarthy and then from RFK — LBJ would have stayed in that race in 1968. Despite his withdrawal from the race, deep down LBJ still had a flicker of hope that the Democratic National Convention would be deadlocked, turn to the outgoing LBJ, draft him into the race, nominate him, and he’d be the conquering hero, vanquishing Nixon and bringing the Vietnam War to an end.
LBJ was also a man of contradictions, though. Throughout his life, he always said that he would die young because all of the men in his family died by the time they were 64 or 65. As much as Johnson was addicted to power and craved the love of the American people (something that he never received like JFK did, which “broke his heart” according to Richard Nixon), he was also deeply worried that another four years in the White House would kill him. Worse yet, he would suffer an incapacitating stroke like Woodrow Wilson. LBJ often had a nightmare where he fell ill like Wilson and was an invalid — a shell of a once-powerful man bedridden or feebly being rolled through the White House in a wheelchair. It was an macabre thing to think about, but it was something that frequently haunted President Johnson, especially because he had suffered a near-fatal massive heart attack in 1955 when he was Senate Majority Leader. The confident, arrogant, impetuous, strong-willed LBJ wanted to take on Nixon and serve four more years in the White House. The sensitive, insecure, depressed LBJ considered resigning, didn’t think he’d live through the next term (1969-1973), and often had to receive a pep talk from Lady Bird to get his act together and go to work. So, with LBJ, it would actually depend on which LBJ you got on decision day when it comes to whether he would have sought a third term if not for the disastrous results of the 1968 New Hampshire Democratic Primary.
By the way, Lyndon Johnson died on January 22, 1973. If he had served a third term, it would have ended on January 20, 1973, just two days prior to the day that he actually died.