South Korean President Moon Jae-in said Sunday that North Korean leader Kim Jong Un had reaffirmed his commitment to “completely denuclearize the Korean Peninsula” and looked forward to meeting President Donald Trump on June 12, in the latest attempt by the two Korean leaders to keep recent engagement efforts on track.

The remarks by Mr. Moon came a day after the two Korean leaders met for an unannounced summit at the inter-Korean truce village of Panmunjom on Saturday, in the second meeting between the two men in as many...