Many observers agreed the North had won this round in the propaganda battle, but many hurdles lie ahead before a lasting solution is reached.
South Korean officials said Moon had been invited to Pyongyang to speak with Kim Jong Un during the talks and lunch that Moon hosted at the presidential Blue House in Seoul on Saturday.
A spokesperson for President Moon said he hoped the two countries would work towards a summit and called on the USA and North Korea to resume dialogue.
The Moon government has been engaging in a delicate balancing act between Pyongyang and its longtime ally Washington after Kim Jong Un offered to send a delegation to the Olympics in the South. Moon has accommodated the North's demands on Olympics participation in hopes of persuading Pyongyang to return to the negotiating table to discuss denuclearization, while trying to allay concern in Washington that the North was using a "charm offensive" to simply ease sanctions and earn time to complete its nuclear weapons program.
Are these Winter Games having the effect South Korea had hoped, in thawing relations with the north?
The communist state has test-fired numerous missiles, including three ICBMs in recent months, and conducted its sixth and most powerful nuclear test September 3, prompting the U.N. Security Council to impose its toughest sanctions yet in a bid to get the North to stop.
The comments represent a departure from the Trump administration's previous stance that it would not engage in any dialogue with North Korea unless Kim Jong Un's regime first agreed to the goal of denuclearization.
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White House officials and other critics believe North Korea is trying to use the worldwide stage to gain sympathy and ease punishing economic sanctions and diplomatic pressure that are beginning to bite.
"What can not be overlooked is that Moon did not directly demand North Korea to abandon its nuclear development".
"Both of us reiterated to each other tonight that we will continue to stand strong and work in a coordinated way to bring maximum economic and diplomatic pressure to bear on North Korea", he said.
South Korea said on Monday it will try to arrange more reunions for families divided by the Korean War and seek to lower military tensions with North Korea as the first steps towards establishing grounds for a rare summit between the two Koreas. He also called for the need of a quick resumption of dialogue between North Korea and the United States. The two did not interact - although Pence, who arrived late, greeted the other dignitaries around him - prompting criticism of the American vice president. He then was stone-faced as he sat to the left of Moon in the VIP box at the opening ceremony while Kim Yo Jong smiled a row behind him, just a few feet away.
The two Koreas are still technically at war after the 1950-53 conflict on the Korean peninsula ended in a ceasefire and not a truce.
Kim Yo Jong, the younger sister of Kim Jong Un, "courteously" handed over a personal letter from the North Korean leader to Moon during talks on Saturday and also told Moon of her brother's "intention", KCNA said, without elaborating on what that was.
Kim's sister is said to be acting as his special envoy in the south during the Games.