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In China it is called 五子棋, I used to play a lot with my friends on paper during class XD

That's the vanilla Gomoku / 五目並べ without limitations for Black. The Renju version is called 連珠 in Japan.

Actually it is not. That is the improved version of it with additional rules to try to avoid black always win.

Oh this is really nice. Sorry I will take a deeper dive.