You're getting confused because the distinction you make between "dynastic names" and "country names" don't exist. Names like Han, Chosŏn, or Qing aren't the names of the dynasty (which would be Liu, Yi, and Aisin Gioro respectively), they're the names of the actual state. This is how they are always referred to in East Asia; as guo hao, the "name of the state."
Consider the very first Chinese empire, the Qin. The Qin weren't called that because that was the name of their dynasty, it was because the empire was founded by the Kingdom of Qin conquering all other Chinese kingdoms. Qin was always the name of the state.
Same with Han, with Tang, with Song. They all refer to regions in China where the founding emperor built up a base.
With Ming and Qing it's a bit different, because these state names are ideological. But these are still not dynastic names, any more than Burkina Faso is the name of the Burkina Fasoan government as opposed to the actual country.
Because in East Asia, a state was associated with a ruling dynasty to a level greater than in Europe, the name of the state changed when the ruling dynasty was deposed. But "Ming" or "Chosŏn" were never dynastic names, even to the extent that Osmanli was in the Ottoman empire. Imagine: in the sixteenth-century Ottoman empire, no Armenian merchant or Egyptian peasant would think of himself as an Osmanli, only as a subject of the Osmanli class (which referred solely to the bureaucrats and officers who served the House of Osman). In the sixteenth-century Ming empire, every Chinese person would think of himself as a "Ming person."
Because the state name "Qing" wasn't exactly popular following the Xinhai Revolution, the revolutionaries used Zhonghua as the name of their state, an antique word referring to Chinese civilization. For a European analogy, imagine that the name "France" was so closely associated with the Ancien Regime that Robespierre changed it to "Gallia."
Because the Yi weren't nearly as unpopular as the Aisin Gioros, and because there was no similar antique word referring to Korea that was palatable to Korean nationalism (most ancient words for Korea are from a Sinocentric perspective), Koreans stuck with Chosŏn and Taehan.