The Oxford English Dictionary says the following. Historically, the comparative of the adverb “well” was “bet” (the link requires subscription) whereas the comparative of the adjective “good” has been “better”. The use of the word “better” as an adverb appeared around the 13th century, and it superseded “bet” by around 1600.
As Kosmonaut writes in a comment on the question, in modern English, the word “better” is used as both an adjective and an adverb. The Merriam-Webster dictionary agrees on this.
The use of "Korea" to mean "South Korea" is far more common in the adjectival form. E.g. "Korean automaker", "Korean pop music" or "Korean soap opera".
Owing to North Korea's status as an economical, cultural, and political walled garden - or perhaps it's more apt to say "walled mudflat" - there are very few words to which the adjective "Korean" commonly applies for outsiders. "Dictator", of course, is the obvious exception.
Thanks largely to the media, this adjectival shorthand usage is firmly entrenched.
In nounal form, my sense is that the mapping is much further from one-to-one. (Since I don't know how to search one of these online corpuses for "this when not neighbored by that", I can't confirm that anecdotal evidence with data, alas.)
While "Korea" would be formally correct within their own language since they are formally the Republic of Korea, in English the pattern of usage I observe is to at the very least differentiate them on first usage.
For example, this is a pattern I hear often in the financial news: "Korean automaker Kia announced today it will be bringing 50,000 new jobs to South Korea. The company has long dominated the auto market in Korea and is now seeking..."
First nounal mention: South Korea. Subsequent mentions can be simply Korea.
So I'd lean toward not linking the tag Korea to South Korea exclusively.
Best Answer
Singular:
Plural:
Do not confuse with:
vs