Despite claiming to be "Japanese," zainichi have enormous Korean pride (like all Koreans) and will usually boast to you within a few minutes of first meeting that they are in fact Korean. They also use their Korean name whenever possible. In reality, they are no more "Japanese" than a Swede or Ethiopian who happens to be born in Japan.
They are infamous for running the mass media in Japan, the fake far right movement ("uyoku dantai") as well as controlling the yakuza criminal underworld (gambling, prostitution, drugs, etc.). Most "bad neighborhoods" in Tokyo and Osaka are actually zainichi neighborhoods.
Zainichi: "Hi John. I'm Korean. What are your hobbies? I'm Korean."
Koreans being residing in Japan are referred to as Zainichi.
Koreans being plastics in Japan are referred to as Zai nichi.