He's probably a speaker of Russian, so he assumes the Russian meaning of the word to be canonical. Which is rather ignorant, because the word has a different meaning in Ukrainian - it literally means 'country'. It is similar in Polish ('country' vs 'edge'/'border'/'region' in Russian) and probably other Slavonic languages.
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.
> The traditional theory (which was widely supported by historians and linguists in the 19–20th centuries, see e.g. Max Vasmer's etymological dictionary of Russian) is that the modern name of the country is derived from the term "ukraina" in the sense ‘borderland, frontier region, marches’ etc. These meanings can be derived from the Proto-Slavic noun *krajь, meaning ‘edge, border’. Contemporary parallels for this are Russian okráina ‘outskirts’ and kraj ‘border district’.
However:
> Some Ukrainian scholars, such as Hryhoriy Pivtorak, Fedir Shevchenko, Mykola Andrusyak, Serhiy Shelukhin believe that the name is derived from ukraina in the sense of ‘region, principality, country’. Many medieval occurrences of the word can be interpreted as having that meaning. In this sense, the word can be associated with contemporary Ukrainian krajina, Belarusian kraina and Russian and Polish kraj, all meaning ‘country’ (see Translations, 'region of land').
So different people have different interpretations.
In general, when linguistics gets politicized, and there are multiple competing theories with various implications, take a look at the geographic spread of each. If it's politically relevant for one country, and one particular theory is only popular in that country, it's probably being promoted for political reasons. If it's widespread in academic circles of many different countries, including those that don't have any stakes in any associated politics disputes, it's more likely to be factual.
It doesn't really. Ukrainian meaning of "u" is different from Russian, it's more like a Russian "v". For example "u lavtsi" translates to something like "in the shop", but never "near the shop", at the same time in Russian it would be "v lavke". It's just that heavily pushed "borderland" narrative makes it hard to think straight about this, especially for Russians.
But we're not talking about either one of those. The word is much older than that - about 8 centuries old if you count back to the first time it can be found in written text, and about 4 centuries if you count to its first use specifically to refer to Ukraine.
I'm not saying that 8 centuries ago, that territory was called by that name. Why would it be, when it was still the heartland of Kievan Rus? I'm saying that the word "оукраина" was used to mean borderlands then, such ones as they had.
And then 4 centuries later, that exact word became the name of the border territory of Polish-Lithuanian Commonwealth that covers pretty much the entirety of today's Western and Central Ukraine, in the language of people inhabiting that territory, said language being a direct descendant of the language spoken 4 centuries before...
I would dare say that a straightforward explanation is that the word retained its meaning, and was used in that meaning, because at that time it really was applicable to the territory in question. All the dots connect here in obvious ways. In contrast, if we assume that "ukraina" was derived from "kraina" for "homeland" instead, and try to backtrack, we need to explain how "u-" got there, and why the fact that the result matched the word that historically meant "borderlands" is entirely coincidental. I've seen some very convoluted explanations of this that could work, but Occam's razor still applies - between two explanations that both match observed evidence, the simpler one is preferable.
Which is probably why the "borderlands" etymology is widely accepted among linguists throughout the world, while various alternative theories all seem to have Ukrainian authors. I get that this is a very politicized question in Ukraine, especially since Russian nationalists and irredentists have been trying to interpret the "borderlands" etymology to imply that Ukraine was historically self-identifying as a border territory of Russia - which is, of course, blatantly false. But I don't think that starting with a premise that is more politically palatable, and then backtracking from there to recreate the (less likely) etymology to fit, is a good way to deal with it.
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.
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.
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.
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.
https://en.wikipedia.org/wiki/Polish%E2%80%93Lithuanian_Comm...
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.
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?
South Slavs have multiple historical examples of Krajina(s) and they all represented bordering area, edge, (military) frontier. In a broad sense kraj/krai means area, land, place. But krajina(s) were always bordering areas.
https://en.wikipedia.org/wiki/Krajina
There's also Polish example of Krajna, the bordering region between Greater Poland and Pomerania.
https://en.wikipedia.org/wiki/Krajna
Of course, linguistics can and do get politicized so...