Adjective or noun when referring to plural citizenship

To use an adjective of nationality, you can simply say

We are Italian [Japanese, American, German, Egyptian, Korean, etc.]

or

I am Italian [Japanese, American, German, Egyptian, Korean, etc.]

The fact that you are an Italian person is implied (and obvious).

If you want to stress the personhood of the individual or group, say

We are Italians [Japanese, Americans, Germans, Egyptians, Koreans, etc.]

or

I am an Italian [a Japanese, an American, a German, an Egyptian, a Korean, etc.]

Note that in cases where the the nationality adjective takes the -ese ending, as in Japanese or Portuguese, you do not add the -s to form the plural.

Finally, there is nothing wrong with saying either Korean or Koreans.

EDIT The -i ending follows the form of the -an ending: Saudi/Saudis, Israeli/Israelis, Pakistani/Pakistanis. And there are some nations for which no really satisfactory plurals or adjectives exist: What, for example, does one do when referring to people from Ivory Coast? Probably call them Cote d'Ivoirian, but that is a bastardization at best.

EDIT 2 in response to OP's EDIT 1

Since it now appears you are talking about any demonym, there is no good rule to apply except this: there is no common logic (and, indeed, perhaps no logic at all). You must learn each case individually. In America we have Chicagoans from Chicago, Angelinos from Los Angeles, New Yorkers from New York, Mainers (or Mainiacs) from Maine, and I myself am proud to be a Masshole from Massachusetts (joke!). Note that most place-names ending in a vowel can, however, take the -an ending in one form or another: Tennesseean, Georgian, Russian, Australian, etc., and in some cases you add an i before the an (Floridian, for example). Still, you still have to contend with Naples => Neapolitan, The Netherlands => Dutch, France => French, and so on.

When there are two possibilities, such as ‘We are Italian’ and ‘We are Italians’, the singular emphasises the collective nature of the group, while the plural emphasises the individuality of each of its members. That facility is not available where the plural is identical with the singular, as in Japanese, Chinese and Taiwanese. The forms ‘Saudies’ and ‘Israelies’ are not found, but ‘Koreans’ is certainly used as the plural of ‘Korean’.

The pluralised version is most useful if there are multiple nationalities present. So you might say "This group is made up of Italians, Koreans and French". You could say "Italian, Korean and French nationals", but that is rather more convoluted, and only appropriate if you are distinguishing specific forms of identity.

The difference between "We are Italian" and "We are Italians" is how you are wanting to identify yourself. the former identifies yourself as a member of the group, wheras the latter implies more that you are an individual, who shares a nationality with the others.