Before the 2018 Winter Olympics began in PyeongChang, South Korea, weak ticket sales plagued the Games amid domestic corporate and political scandals and concerns about escalating tensions with North Korea.
In addition to Moon and Kim's sister Kim Yo Jong, the game was attended by Kim Yong Nam, North Korean titular head of state, and International Olympic Committee (IOC) President Thomas Bach.
It is now unclear how the unidentified Korean man was able to jump the barriers and get onto the ice chute but local police in Pyeongchang, the South Korean resort where the two week tournament is held, say they are investigating.
According to the spokesman, Moon replied, "Let's create the conditions and hold the summit".
The South Korean president also stressed the need for the communist North to resume its dialogue with the United States.
US Vice President Mike Pence and his wife were seated beside the Moons and next to Japanese Prime Minister Shinzo Abe and his wife.
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Pence said he would seek to counter North Korea's attempt to use the Olympics for propaganda and invited the father of Otto Warmbier, an American student who died a year ago after being imprisoned in North Korea for 17 months, to the Games in Pyeongchang.
"And Dennis Rodman has confirmed this and he's met both of them personally", he added, referring to the former National Basketball Association player who has met with Kim multiple times. A senior White House official described Pence's tact as ignoring the North Koreans, but not avoiding them.
Pence said his team had not requested a meeting, but that if it did happen, he would continue his message that North Korea must entirely abandon its nuclear and ballistic missile efforts and will remain under pressure until it does so.
North Korean media slammed Pence what they described as "snobbish" behavior.
The UN spokesman said Guterres did not meet with Kim's sister.
North Korea has also been blamed for a series of cyber incidents including the WannaCry global ransomware attack, which infected 300 000 computers worldwide last May.
Moon sat in the same row as Kim and North Korea's ceremonial leader, Kim Yong-nam, to watch the unified Korean team's match against Switzerland at Kwandong Hockey Centre in Gangneung.
USA's J.R. Celski competes in the men's 1,500m short track speed skating heat event during the Pyeongchang 2018 Winter Olympic Games on February 10, 2018.