It has become an awkward ritual of the modern presidency that the current occupant of the Oval Office is called upon to deliver a generous historical judgment of the previous one. With the opening of each new presidential library, the members of the world’s most exclusive fraternity put aside partisan differences to honor the shared experience of running the nation in difficult times.

The task in such moments is especially acute when, as with Mr. Obama and Mr. Bush, the current and previous presidents come from opposing parties and such different sensibilities. The incumbent must hit grace notes without appearing inconsistent with past criticism or, worse, hypocritical. Sometimes a president goes through the motions, dutifully reading what aides put before him. Other times, library dedications have become bonding moments when presidents genuinely grow closer…

Kenneth Khachigian, a chief speechwriter for President Ronald Reagan, said there was a formula: “If you’re a speechwriter, you can write a really nice speech about how Americans can fight and disagree and debate, but at the end of the day we’re all Americans and we have the same goals, and I’ve learned that anybody who sits in this office really at the end of the day has the best interest of everyone at heart.”

For Mr. Bush, 66, who spent the last four years listening in silence as his successor faulted his leadership, this was an emotional day. Bathed in the admiration of a friendly crowd and enjoying a modest resurgence of public appreciation in recent surveys, he choked up as he finished speaking and wiped tears from his eyes after sitting down.

While critics have fumed about what they called the whitewashing of his record in the media blitz leading up to the library dedication, many Americans have been reminded about aspects of Mr. Bush they once liked. Advisers said they hoped the moment would help history draw a fuller picture.

“It’s never been about presenting him,” said Nicolle Wallace, who worked on Mr. Bush’s re-election campaign and in the White House. “It’s been about revealing him.”

Obama chose to focus on a powerful symbolic moment—“the incredible strength and resolve that came through that bullhorn as he stood amid the rubble and the ruins of Ground Zero, promising to deliver justice to those who had sought to destroy our way of life.” It was one of the many artful ways he circumvented mention of a war he called “dumb” before it had even started.

The Democratic speeches were short, which was helpful, given the huge chunks of Bush’s presidency they have sharply critiqued and, in Obama’s case, run against twice. All three congratulated Bush on the President’s Emergency Plan for AIDS Relief that has saved so many lives. Carter also thanked him for keeping a promise to help resolve a conflict in Sudan. Clinton, who handed Bush a nation with a hard-won budget surplus and saw it fall into fiscal disarray, said he’d limit himself to areas “beyond controversy”—and mentioned, in addition to AIDS, Bush’s immigration efforts and post-presidential humanitarian work…

The most expansive comments about Bush were those Obama and Clinton offered about his personality: unpretentious, direct, and disarming. “To know the man is to like the man,” Obama said.

“We expanded freedom at home by raising standards in schools and lowering taxes for everybody. We liberated nations from dictatorship and freed people from AIDS,” he said.

And he added, alluding to the Sept. 11, 2001 attacks, “when our nation came under attack we made the tough decisions required to keep the American people safe,” he said.

“My deepest conviction – the guiding principle of the administration – is that the United States of America must strive to expand the reach of freedom,” he declared. Freedom, he said, “sustains dissidents bound by chains, believers huddled in underground churches and voters who risk their lives to cast their ballot.”