A common term of full-contact is 直接打撃制.
フルコンタクト is also used. In fact you can see フルコンタクト in bracket to explain 直接打撃制.
More often we hear people say フルコンタクト空手/直接打撃制空手 rather than フルコンタクト組手.
直接打撃制ルール/フルコンタクトルール are also used.
フルコンタクト組手 is correct but less often heard in Full Contact styles. To separate the different styles of Kumite in one organisation using フルコンタクト組手 is okay.
As for no-contact Karate more often we say 伝統空手 or 寸止め空手.
As for those other styles I have no information since I am interested in Kyokushin only.