“From Chinese” and “From Korean” are more accurate at least. It would still feel a bit odd to me as a user, unless names in general are annotated with similar etymologies. But that’s a design decision for the application developer to make. There are certainly uses for that sort of presentation. A biblical scholar might prefer a specialized map of the Holy Land blanketed with annotations comparing the classical and modern names of places, as a sort of critical apparatus.
Maps for a more general audience typically make country-based distinctions on features of international interest, but not always explicitly. For example, most English-language maps label this territory as “Falkland Islands (Islas Malvinas)”. Even if the map generally only labels places in English, it will pointedly include “Islas Malvinas”, the Spanish-language name preferred by Argentina, for diplomatic reasons. For this purpose, name:es=* and perhaps int_name:AR=* would be more informative than name:en-t-es=*. (Since any discrepancy in an international or official name would be a matter of diplomacy rather than linguistics, should we allow int_name=* and official_name=* to be qualified by ISO 3166 country codes instead of IETF language tags?)
For what it’s worth, New Zealand’s English speakers often hybridize the names of features significant to Māori heritage, blurring the line between an “English” name and a name used by English-speakers:
Taking your example at face value, I understand that Ukrainian speakers consider the Māori- and English-derived names to be equally valid but never say both names in tandem. This is a situation that I would’ve used a semicolon-delimited name:uk=* tag for. Perhaps your suggested name:uk-t-mi=* and name:uk-t-en=* tags would clarify the situation further, but it would make more sense if there’s a phenomenon of Ukrainian speakers systematically preferring one source of Ukrainian names over the other, rather than a one-off exception for Aoraki/Mount Cook.
In practice, American English speakers don’t tend to use that particular hybrid name. Few of us have any proficiency in Māori, so the hybrid names seem more unwieldy to us. However, New Zealand’s practice of hybrid naming has started to enter official usage here, with some parks’ English names being changed to incorporate local indigenous names. These indigenous names aren’t necessarily exactly equivalent. For example, Máyyan 'Ooyákma – Coyote Ridge Open Space Preserve is the English name of the park, full stop. Máyyan 'Ooyákma comes from the Chochenyo language, but when the local Muwekma Ohlone people say this name, they’re most likely referring to Coyote Ridge, not the open space preserve that surrounds it. Meanwhile, many local English speakers still say “Coyote Ridge Open Space Preserve” as shorthand. Should there be a name:en-t-en=* for that shorthand, or would some *_name=* variant be good enough?