Moon's office said the president's plane left Thursday after the two leaders visited the sacred Mount Paektu on the North Korean-Chinese border.
To hurry the denuclearization process, Kim said he wants Secretary of State Mike Pompeo to visit Pyongyang and to set up another summit with Trump "as soon as possible", according to a summary by the Pyongyang Press Corps.
"On the basis of these important commitments, the United States is prepared to engage immediately in negotiations", Pompeo said in a statement. Compared to the vague language of their two earlier summits, Kim and Moon seem to have agreed on an ambitious program meant to tackle soaring tensions previous year that had many fearing war as the North tested a string of increasingly powerful weapons.
Seoul aims to jointly announce with the United States an end to the war within this year, a measure Moon said he would discuss with Trump when they meet next week at the United Nations General Assembly in NY. He didn't indicate in his brief remarks whether the USA would be willing to take further steps to encourage North Korean action on denuclearization.
Moon told reporters that he will carry a private message from Kim to Trump about nuclear issues.
Nauert did not respond when asked if the United States was willing to take "corresponding measures", except to say: "Nothing can happen in the absence of denuclearization; denuclearization has to come first".
As Channel News Asia reports, "Mount Paektu is the mythical origin of the Korean people, featured in South Korea's national anthem and various North Korean propaganda". North Korea wants South Korea and the United States to declare the war to be formally over, as a sign that hostilities have ended and to build trust.More news: Ronaldo To Miss Old Trafford Reunion Following Juve Red
Speaking ahead of Mr Pompeo's announcement, Mr Trump welcomed Mr Kim's pledges, calling them part of "tremendous progress" with Pyongyang on a number of fronts, and hailing "very good news" from the summit between the Koreas.
On Wednesday, Trump tweeted about North Korea's commitment to denuclearization, saying Kim has agreed to "nuclear inspections" and to "permanently dismantle a test site and launch pad".
The words "observation" and "permanent dismantlement", which the North used in the Pyongyang Declaration, ultimately mean the same thing as "verifiable and irreversible dismantlement", Moon said.
Agreeing to a declaration on ending the Korean War before North Korea has reduced its offensive threat - including the thousands of artillery guns it has trained on USA ally South Korea - is opposed by many who fear that the North is maneuvering to win a withdrawal of the almost 30,000 US forces in the South.
Mr Moon was born in South Korea in 1953 during the Korean War.
The South Korean leader also said he'll convey to Trump his and Kim's desire to get a declaration ending the Korean War by the end of this year.
Asked about Pompeo's statement on Wednesday welcoming plans for the dismantlement of all facilities at Yongbyon in the presence of US and IAEA inspectors, Nauert said Moon and Kim had talked about inspectors. He said, "The era of no war has started". Moon, who was the handmaiden of the Kim-Trump dialogue back in the spring, was under considerable pressure this week to extract concessions from Kim that could sustain the rapprochement between Washington and Pyongyang. Kim reportedly told South Korean diplomats at a meeting in early September that his aim is to denuclearize by the end of Trump's first term in 2021.