I have hardly called gaijin or foreigner when I describe about people from other countries. In my opinion, there is no difference between the two words.
Does the word “gaijin” sound exclusive whereas “foreigner” doesn’t sound exclusive? It does sound exclusive also. Gaijin just means foreigner in Japanese. The nuance is also totally same.
To me, both words sound a sort of cold and unfriendly and because of this reason I tend to abstain from using them and choose another word instead.
The better version of gaijin or foreigner might be “international person”, “non-japanese people”, or just “people from a different country”. I recommend that you guys use these words.