On February 23, 1897, the Independent published this article about Seoul's changing skyline:
What's interesting about that article is that one gets the idea that a 'Europeanized' 'foreign settlement' with 'European buildings' was an accoutrement that Seoul - and Korea- needed in order to be truly modern.
As for the buildings referenced, the French Legation was located on what is now the grounds of Ewha Girls' High School in Jeong-dong. The 'new Cathedral in Chonghen (?)' refers, of course, to Myeong-dong Cathedral, while "another Catholic Church in Yakhen" is Yakhyeon Catholic Church, near Seoul Station.
The Russian legation once looked like this, but now only the tower stands (see here).
The British Legation still stands today, while the long-gone customs building stood near the old Shin-a Ilbo Building, according to a nearby plaque. The "New Methodist Church in Chongdong" still stands today as Chungdong First Methodist Church, while a later building from the 'Methodist School' [Paichai Hakdang] still exists today as the Appenzeller/Noble Museum (the building standing at the time of the Independent article can be seen here). The Japanese consulate (which was built in 1895 and stood on the northern slope of Namsan) can be seen here (fourth photo).
I've written about the Jeong-dong area before. Though Seoul may have been modernizing, neither that, nor the declaration that Korea was an empire later in 1897, saved Korea from being colonized by Japan. An interesting book being published this spring is Todd Henry's Assimilating Seoul: Japanese Rule and the Politics of Public Space in Colonial Korea, 1910-1945; an excerpt of the book can be read at the link. I imagine the book will incorporate some of his earlier writings about colonial Seoul, including "Respatializing Choson's Royal Capital: The Politics of Japanese Urban Reforms in Early Colonial Seoul, 1905-1919," a chapter in the book Sitings: Critical Approaches to Korean Geography. This chapter has a fascinating map showing the Japanese government's original plans for colonial Seoul, which included streets radiating out from a plaza along present-day Euljiro. The plan was never carried through, but one city in which such a plan was carried out is Jinhae; a 1946 map of the city can be found here (first result). Right next to Jinhae is Masan (actually, both cities were incorporated into Changwon a few years ago); here is a post about Masan's first modern buildings.