Learn English – What word do I use to describe people from India & neighboring countries

You might be able to use the term "East Asians". Of course, you have to consider whether your audience is sufficiently familiar with the East Asia region or not. Also, Mongolia is at times included as an East Asian country; and to further muddle the boundaries, some people use the term "East Asia" to also include Southeast Asia (not a practice I personally approve of).

In order to avoid any ambiguity you could define the term when you first use it in an article, e.g. East Asians (i.e. people from China, Japan, and the Koreas).

As for "Oriental", I don't believe it carries a negative connotation in most of Europe (but I'm not 100% sure on this). To me, the term carries connotations of romanticism and adventure - probably because of the Orient Express. However, it is definitely an antiquated term and on that basis alone I would recommend you not to use it. The fact that some people (possibly a large amount of people) take offence to the term is another good reason to phrase it differently. Note that both Oxford Dictionaries Online and Merriam-Webster list "Oriental" as having negative connotations. (Also note that they disagree somewhat on the geographical scope of the term, with the former dictionary interpreting it as referring in particular to East Asians and the latter as referring to Asians in general.)

And what do you mean by people from "this part of the world"? Are you referring to actual Chinese, Japanese and Korean citizens or to people who can be ethnically classified as such but are citizens of countries outside of East Asia? I can tell you that my Chinese friends (Chinese nationals) aren't even familiar with the term "oriental" and that they would refer to themselves as Chinese (and that makes perfect sense as you would most likely also refer to yourself as American rather than North American if someone asked). If they'd use a regional term they would definitely go with "Asian" - but as you already noted, this term is too broad to be of much use in many cases.

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.