OSAKA, Japan — They were having a good time. Like old friends reuniting, they warmly shook hands, smiled and chatted amiably. And then President Trump brushed off Russia’s interference in American democracy with a joke as President Vladimir V. Putin chuckled.
The first encounter between Mr. Trump and Mr. Putin since the special counsel Robert S. Mueller III reported that Russia conducted a “sweeping and systematic” operation to sway the 2016 election proved more convivial than confrontational. Rather than challenge Mr. Putin, Mr. Trump treated it as a laughing matter.
In the process, he triggered a fresh furor over his accommodating approach to Russia and brought back old questions that have haunted him since he took office: Angry at perceived challenges to his legitimacy, he has long dismissed the conclusions of American intelligence agencies that Russia sought to help his campaign.
But while Mr. Trump once hoped to leave the investigation behind and finally recalibrate the Russian-American relationship, he instead put the issue back in the spotlight as House Democrats prepare to question Mr. Mueller on camera next month.
As he sat down on Friday with Mr. Putin on the sidelines of an international summit in Japan, Mr. Trump was asked by a reporter if he would tell Russia not to meddle in American elections.
“Yes, of course I will,” Mr. Trump said.
Turning to Mr. Putin, he said, with a half-grin on his face and mock seriousness in his voice, “Don’t meddle in the election, President.”
As Mr. Putin smiled and tittered, Mr. Trump pointed at another Russian official in a playful way and repeated, “Don’t meddle in the election.”
The levity came at a time when Mr. Putin has felt emboldened on the world stage, flexing Russian muscle in Europe, Asia, the Middle East and even South America. In an interview published just hours before the meeting, Mr. Putin celebrated the rise of the populist right in Europe and the United States and declared that traditional Western-style liberalism “has become obsolete.”
Mr. Trump did not dispute Mr. Putin’s view but seemed almost to share it. As reporters and photographers entered their meeting room to set up cameras and microphones on Friday, the American president offered the sort of disdain for journalists sure to resonate with an authoritarian like Mr. Putin.
“Get rid of them,” Mr. Trump said. “Fake news is a great term, isn’t it? You don’t have this problem in Russia, but we do.”
“We also have,” Mr. Putin insisted in English. “It’s the same.”
In fact, Mr. Putin has made a hallmark of his nearly two decades in power a takeover of major news outlets. Russia’s relatively few independent journalists often come under intense pressure and, in some cases, have even been killed.
It fell to other leaders gathered in Osaka, Japan, for the annual Group of 20 summit meeting to volunteer the rebuttal to Mr. Putin’s worldview that Mr. Trump did not.
“What I find really obsolete are: authoritarianism, personality cults, the rule of oligarchs,” said Donald Tusk, the president of the European Council. “Even if sometimes they may seem effective.”
The bonhomie between Mr. Trump and Mr. Putin came in sharp contrast to Mr. Putin’s frigid meeting with Prime Minister Theresa May of Britain, their first since a former Russian spy living in her country was poisoned by agents that Britain has traced to Russia. Stiff and severe, Mrs. May refused to smile or exchange pleasantries as she sat down with Mr. Putin. Aides later said she upbraided him behind closed doors over the poisoning, calling it a “truly despicable act.”
Mr. Trump’s friendlier session touched off another domestic backlash like the one he endured after their last official meeting in Helsinki, Finland, last year when, standing at Mr. Putin’s side, the president challenged the conclusion of his own intelligence agencies about the Russian election operation and credited the Kremlin leader’s “extremely strong and powerful” denial.
“As Robert Mueller said, Russian interference in our democracy should concern every American,” Representative Adam Schiff, a California Democrat and chairman of the House Intelligence Committee, wrote on Twitter on Friday. “But not the president, apparently, who thinks it’s a joke. And if we don’t act to secure 2020 against further interference, Putin will have the last laugh.”
Former President Jimmy Carter, who at times has been sympathetic to Mr. Trump’s complaints about media coverage, responded sharply on Friday to the president’s comments in Osaka. Going further than some Democrats, he even suggested that the president did not genuinely earn the office.
“I think a full investigation would show that Trump didn’t actually win the election in 2016,” Mr. Carter said at a conference sponsored by the Carter Center, in Atlanta. “He lost the election and he was put into office because the Russians interfered on his behalf.”
That assessment goes to the heart of Mr. Trump’s resistance to taking the Russian interference more seriously, according to his advisers. In his view, the intense focus on the matter is mainly a partisan effort to undermine his legitimacy as president.
And he has argued that there was nothing wrong about accepting incriminating information about an election opponent from a hostile foreign power, saying recently that “I’d take it” and did not necessarily see a need to call the F.B.I.
Moreover, Mr. Trump maintains that whatever happened in 2016, it is in the interest of the United States in a dangerous world to have a more constructive relationship with Russia.
“It’s a great honor to be with President Putin,” Mr. Trump said as they sat down. “We’ve had great meetings,” he added. “We have had a very, very good relationship. And we look forward to spending some pretty good time together. A lot of very positive things going to come out of the relationship.”
Mr. Putin said they would discuss trade, disarmament and other issues. “All this will be built on a very good relationship that will be between us,” he said. “I think that the results of this meeting will be excellent.”
Mr. Trump, who has withdrawn the United States from the Intermediate-Range Nuclear Forces Treaty of 1987, and whose national security adviser has signaled that he would not extend the New Start treaty signed with Russia in 2010, has been interested in negotiating a three-way arms pact with Russia and China.
Mr. Putin has not ruled out such an accord, but he has noted that China has only a fraction of the nuclear weapons that the United States and Russia have. Sergey V. Lavrov, the foreign minister, welcomed Mr. Trump’s outreach. “We are ready for cooperation to the extent the States are ready to get involved in it,” he said.
The White House summary of the leaders’ meeting indicated that they talked about such an accord as well as Iran, Syria, Venezuela and Ukraine. The summary made no mention of election interference, nor anything about two Americans who have been arrested by the Russian authorities on disputed charges.
Likewise, it said nothing about an international investigation this month that pointed to Russia in the 2014 downing of Malaysia Airlines Flight 17 over Ukraine, which killed all 298 people on board. International prosecutors have indicted three men with ties to Russian military and intelligence agencies in the destruction of the passenger jet and implicated, without charging, a senior aide to Mr. Putin.
Nor did the summary indicate that the leaders talked about Russia’s seizure of three Ukrainian ships and two dozen sailors last November, events that prompted Mr. Trump to cancel a scheduled meeting with Mr. Putin, and that remain unresolved. When a reporter asked about the ships and sailors on Friday, the president said, “We haven’t discussed them.”
While Mr. Putin did not address the election issue with reporters on Friday, he scoffed at the idea of Russian involvement in an interview before flying to Osaka. He advanced the same line of argument that Mr. Trump does: that he won in 2016 because he was in touch with Americans.
“Russia has been accused, and, strange as it may seem, it is still being accused, despite the Mueller report, of mythical interference in the U.S. election,” Mr. Putin told The Financial Times. “What happened in reality? Mr. Trump looked into his opponents’ attitude to him and saw changes in American society, and he took advantage of this.”
He complimented Mr. Trump’s political skill. “I do not accept many of his methods when it comes to addressing problems,” Mr. Putin said. “But do you know what I think? I think that he is a talented person. He knows very well what his voters expect from him.”
Following the interview, however, the Kremlin pulled back Mr. Putin’s rejection of liberalism. “Putin is still very close to the ideas of liberalism,” Dmitri S. Peskov, his spokesman, told reporters, but added that it cannot be forced on others.
“We agree completely that authoritarianism and the rule of oligarchs are obsolete,” Mr. Peskov said. “This cannot be an effective system. At the same time, if authoritarianism exists somewhere, this is a question of the people of these countries. We should not judge them and change the regime and government in these countries.”