No, I'm using the historical meaning of "u", from before Russian and Ukrainian were even distinct languages. Here's the "Tale of Igor's Campaign", 12th century, in the contemporary East Slavic:
Игорь ждетъ мила брата Всеволода.
И рече ему буй туръ Всеволодъ:
"Одинъ братъ,
одинъ светъ светлый -
ты, Игорю!
оба есве Святъславличя!
Седлай, брате,
свои бръзыи комони,
а мои ти готови,
оседлани *у* Курьска напереди.
Note last line. English translation:
Igor awaits his beloved brother,
and Fierce Bull Vsevolod speaks:
“You are my one brother,
Igor,
my one shining light;
we are both sons of Sviatoslav.
Saddle, brother,
your swift horses,
for mine are ready,
saddled ahead *at* Kursk.
Ukrainian translation:
Дожидає Ігор брата Всеволода.
Каже йому буй-тур Всеволод:
«Один брат,
один світ світлий,
Ігорю,
Обидва ми Святославовичі.
Сідлай, брате,
свої коні бистрі,
Мої-бо вже готові стоять,
*Під* Курськом осідлані.
The word "ukraina" was first attested at roughly the same time these lines were written.
No offence but pretending that google translate is a proper way to evaluate translations of almost 1000 year old text is beyond funny. It's obviously before Russian lang. existed since Moscow was founded 400 years after this text was written. At this time Kiev was not under any external power so you theory of the origin of the name is also very questionable.
First of all, where do you see any Google Translate above? Both are professional translations of the original. And the first snippet isn't Russian - it's Old East Slavic (from which both Russian and Ukrainian eventually came).
And of course I'm not claiming that the word "ukraina" was used to refer to what today is the country Ukraine back in 1200s - that would be ridiculous, for exactly the reasons you cite above. No, my point rather is that the word "ukraina" - "оукраина" - is attested in documents from that period to refer to areas that they (the Kievan Rus state) considered borderlands, and has fairly obvious and transparent etymology of "u+kraina", both parts of which are also attested in written sources.. It is solely about the etymological origin of the word - that, yes, it does in fact mean "borderlands" historically - not about the origin of the name "Ukraine" for the country. That came much later.
As for the latter, it shows up first somewhere circa 1600s, and it first appears in Polish sources (as self-designation of people inhabiting the territory). At that point, the entire Western and Central Ukraine was a part of the Polish-Lithuanian Commonwealth - and for them, it was definitely "borderlands" in all senses. Geographically it was the easternmost territory of the state. Culturally it was significantly different from Polish heartland - language, religion etc - and further east and south there were culturally alien Turkic (Tatar) tribes. Militarily, it was a buffer zone between the Commonwealth and the Tatars, and to some extent with Russia, with constant low-key hostilities.
It's exactly the kind of area that would be called e.g. "the Mark" in German-speaking parts of Europe. Indeed, when the revived Polish state later occupied large parts of Western Ukraine after WW1, they referred to it as "kresy wschodnie" in Polish, literally meaning "eastern borderlands" - old habits die hard.
The first written mention is Kiev litopis that was way before the polish source, and it would be pretty pointless for Kievans to refer to themselves as borderland. I guess given the politicization of debate it's hard to find any objective info. From point of view of western neighbors it is possible that your interpretation makes sense but it does not explain why Kiev's litopys would use this name. On the translation some of words that have actual meaning are just transliterated which gave me the impression that it was google translation.
Again, the first written mention of what - the word "ukraina", or the use of that word to refer the territory that's Ukraine today? I tried to draw that distinction as clearly as I could above.
So far as I know, all uses of the word "ukraina" in Kievan sources do not refer to the territory as a whole, nor to territories near Kiev, but rather those that were borderlands with respect to Kiev as the capital (e.g. Halych). Of course, I may well be missing something - can you give an example of a Kievan source that use "ukraina" to refer to the country as a whole, or to its heartland?
It's debated whether or not it referred to territory as whole with most historians not supporting that idea, yet the territories it referred to were pretty much next to Kiev so the name borderland again very debatable.
