Moon and Kim resumed their talks on Wednesday at around 10 a.m. (0100 GMT), this time without any other attendees.
Kim pledged to work toward the "complete denuclearization of the Korean peninsula" during his first encounter with Moon, and at his summit with Trump in June.
Moon responded by expressing his own thanks to Kim for making a "bold decision" in a New Year's speech to open a new era of detente and send a delegation to the South Korean Winter Olympics in February.
North Korea first staged its mass games in 2002, when Kim's father, Kim Jong Il, was leader.
Earlier, the leaders paraded down the streets of Pyongyang in Kim's black Mercedes limousine to loud cheers from almost 100,000 North Koreans who waved flowers and chanted "Motherland!Unification!"
This week's summit is meant to craft concrete steps to implement the Panmunjom Declaration, named after the border village where they first met, Seoul officials said. "I've been waiting and waiting for today".
During their motor parade through Pyongyang's landmark Ryomyong Street, a new residential district launched previous year under Kim's initiative to modernize the city, Kim and Moon briefly stepped out of the vehicle to greet and take flowers from members of the crowd. Talmadge is the AP's Pyongyang bureau chief.
But discussions over how to implement the vague commitments have since faltered, with Washington demanding concrete action towards denuclearization by North Korea before agreeing to a key goal of Pyongyang - declaring an end to the war.
North and South Korea agreed that the Korean Peninsula should turn into a "land of peace without nuclear weapons and nuclear threats", Moon said.
The conflict ended with an armistice, not a peace treaty, leaving US -led United Nations forces including South Korea technically still at war with the North.
Kim gave the South Korean president an exceedingly warm welcome on Tuesday, the first day of the summit, meeting him and his wife at Pyongyang's airport - itself a very unusual gesture - then riding into town with Moon in an open limousine through streets lined with crowds of North Koreans, who cheered and waved the flag of their country and a blue-and-white flag that symbolizes Korean unity.
USA officials involved in the latest negotiations have said North Korea has refused to even start discussions about defining denuclearization. The talks are expected to be followed by a joint statement and a news conference, Moon's office said. Moon will return home early Thursday.
Travelling with Moon in North Korea are South Korean business tycoons, who met North Korean Deputy Prime Minister Ri Ryong Nam, who is in charge of economic affairs on Tuesday.
"If and when" they reach an agreement, they would announce it jointly but not take questions, he said.
The institution was sanctioned by the U.N. Security Council a year ago as part of global efforts to rein in Pyongyang's nuclear and missile programs by drying up its revenue sources.