Remember that these are terms being translated from Japanese/Chinese/Korean into English and vice-versa. The simple answer is that the moniker "empire" comes from Western languages, contexts, and experiences, while in Classical Chinese there is no equivalent term for "empire." The modern Chinese and Japanese term for "empire," diguo / teikoku 帝國, is a nineteenth-century invention that was created from exposure to Western political ideas. I would further guess (though I can't be certain) that this term was invented by Japanese Meiji modernizers using individual Classical characters and "re-introduced" to China, which was the case for many Chinese and Japanese vocabulary terms for imported Western ideas.

As for why Japan was called the "Empire of Japan": the term itself, Dai Nippon Teikoku 大日本帝國, came out of the Meiji Restoration, i.e. when Western political ideas were becoming increasingly popular and political trends in Japan tended towards modernization along Western lines. In contrast, the term "Great Qing" for China's Qing dynasty, Da Qing 大清, originated in the early/mid seventeenth century, long before Western political ideas and norms began to influence the self-identity of polities in East Asia. The descriptor da 大, "great", was historically a common term attached to the dynastic name: Da Ming 大明, Da Yuan 大元, Da Song 大宋, etc., intended to inflect greatness and imply hegemony over continental East Asia and the Sinosphere.

Disclaimer: I have no idea how much of this applies to Korea, but I suspect the broad outlines are applicable.