FWIW, since the word "Ukraine" is first attested in written form as early as 12th century, its etymology has to be derived from the (then still common) Eastern Slavic of that period, rather than the modern meanings of either Russian or Ukrainian. And in that language, "ukraina" - "оукраина" - did mean "borderlands"

BTW, it doesn't appear to have become the name of the territory, and later the country, because it was borderlands of Muscovy/Russia. Rather, it seems to have originated in the western parts, because they were the borderlands of Polish-Lithuanian Commonwealth. Russia historically preferred the term "Little Russia" (Malorossiya) to imply cultural affinity and justify territorial claims, to the point of proscribing the usage of "Ukraine" in imperial times.