Thoughts on the Origins
Given the fact that there is little information on the press itself other than the fact that it's from Japan, I decided to make my own theories and ideas about the origins of the printing press.
From all of the research I've done on the region and history around the time period that the press was in Korea, I have reason to believe that the press originally came from Europe as Christian missionaries came to Japan. However, I don't think I can pinpoint exactly which country the press might have come from as many Western powers came to Japan. As Christian missionaries were spreading their religion in Japan, they also brought over their knowledge in science, politics, and technology. One of the things they might have brought to Japan was this specific printing press.
Another way that the press might have been in Japan is that the Christian missionaries brought over information about printing presses to Japan. The Japanese people could have learned how to build Western printing presses and created their own, rather than the West bringing the printing press.
Either way, Japan and Korea signed the Treaty of 1876, which made it so Japan could have diplomats in and relations with Korea. As Japanese diplomats could enter Korea, it would make sense that Japan and Korea exchanged information and technologies. From this treaty, Japan would have been able to bring the printing press to Korea to be used. Specifically, Yeong-Hyo Park, an official emissary, brought over the printing press to help modernize Korea (Kim 805).