Skeptics of how much the meeting achieved pointed to the North Korean leadership's long-held view that nuclear weapons are a bulwark against what it fears are US plans to overthrow it and unite the Korean peninsula.
Trump struck an optimistic tone about his talks with Kim.
North Korea's tightly controlled state media has enthusiastically reported the summit as a great achievement for the country, which remains isolated and heavily sanctioned. 'Will he shake the hand of peace and enjoy prosperity like he has never seen?
Mr Trump announced he would halt "very provocative" and expensive regular military exercises that the U.S. holds with South Korea.
But there has been scepticism from both Democrats and Republicans about the lack of detail and surprise announcement about an end to joint drills with South Korea.
Moon said South Korea would carefully consider joint military drills with the United States and he asked his officials to cooperate with the United States on the issue, his office said in a statement. "I am equally confident they understand that there will be in-depth verification".
"True although it was light on details, I think this represents an overall positiveness in North Korea-U.S. relations which hadn't existed before".
U.S. Secretary of State Mike Pompeo, South Korean Foreign Minister Kang Kyung-wha and Japan's Foreign Minister Taro Kono pose during a trilateral meeting at the Foreign Ministry in Seoul, South Korea June 14, 2018.
By claiming that the North Korean nuclear threat has disappeared, Trump also risks weakening the USA negotiating position in talks on denuclearization that were mandated by the summit and will be led from the U.S. side by Secretary of State Mike Pompeo.
Before showing the two signing their joint statement, the newscaster said Trump made a point of giving Kim a look at his armoured Cadillac limousine, and noted that it is known to Americans as "the Beast".
China's rise has been a long time coming.
Trump described his meeting with Kim as "an interesting and very positive experience" and said North Korea "has great potential for the future".
"Given that they were able to sign this agreement, it will prove that USA remains to be a superpower and also this will give them the right, more leverage on how they will engage with North Korea diplomatically or economically", he said.
Now, when questioned about the human rights abuses in North Korea, the President has defended "tough guy" Kim. "No longer - sleep well tonight!"
Under the deal, the North retains its nuclear warheads, the missiles to launch them and has not agreed to any specific process to get rid of them. It wasn't until Trump's tenure that North Korea began testing an intercontinental ballistic missile and the bellicose rhetoric between the two leaders ramped up.
He said he is certain Kim is honest in his desire to go nuclear-free and to usher in peace and prosperity for North Koreans.
A good recent example of Trump's hot-and-cold temper came the day before the Kim summit. USA officials said it was unclear what types of training involving US and South Korean troops might cross into Trump's now forbidden zone of "war games".
'I just tried to answer the emails and try to get out in front and say, "I had nothing to do with this"'.
The post-summit transformation of North Korea's official version of Trump, who's now being shown by state media looking serious and nearly regal, underscores the carefully choreographed reality show the government has had to perform to keep its people, taught from childhood to hate and distrust the "American imperialists", ideologically on board with the tectonic shifts underway in their country's relationship with Washington.
In his post-summit press conference, Trump made the surprise announcement that the USA would halt joint military exercises with its security ally Seoul - a concession Pyongyang has sought for decades.
Trump also is wrong in saying "nothing happened" in return.
Trump has also been slow to condemn other authoritarian leaders, including President Vladimir Putin of Russian Federation. We should feel thankful towards South Koreans in general and also especially moon Jae-in for his consistent wise and skilled diplomacy. The next scheduled major exercise, involving tens of thousands of troops, normally is held in August.