The new South Korean government has decided not to have a separate name.
An official of the presidential office said Thursday that the government will not use names such as “participatory government” used by the Roh Moo-hyun administration or “government of the people” used during the Kim Dae-jung administration.
The official said that the media could use the terms “Democratic Party government” or “the Moon Jae-in government.” He asked that the media refer to the new government at their own discretion. 
Shortly after he took office, President Moon Jae-in had asked that the government be called the “Democratic Party government” as a way to hold the party in respect. But in a ceremony on Thursday marking the 37th anniversary of the May 18th Gwangju Democratization Movement, the president used the term “Moon Jae-in government” twice during his speech.
Some observers speculate that such usage suggests that the new government has named itself "the Moon Jae-in government.”