SINGAPORE — When President Trump declared that he did not really need to prepare for his legacy-defining meeting with North Korea’s leader, he drew sighs or snickers from veterans of past negotiations. But he had a point: In his own unorthodox way, Mr. Trump has been preparing for this encounter his entire adult life. For an American leader who came of age in the early 1960s, when the United States and the Soviet Union stepped to the brink of nuclear annihilation, the meeting with Kim Jong-un strikes a personal chord, offering Mr. Trump a historic chance to rid the world,...