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I've played or observed a handful of games involving some sort of hand/rhythm/chanting component. Crucially they all have no equipment/items needed to play.
Examples:
Is there a general term for these sorts of games? Are there any particularly well know examples? Are these games known to come from elsewhere, or do many originate in Korea?
Answer: There is no general term.
As Chase Choi said, many of the no-equipment games falls under the category of sulgeim, but it is not an exact match. Kawibawibo, for example, is not a sulgeim. Conversely, sulgame includes a few games where a sort of "equipment" is required, such as 병뚜껑치기.
The term itself is also problematic. Many "sulgames" such as 아이엠 그라운드 is frequently played by minors, including elementary school students, with no alcohol involved.
I have no idea about their origination.
술게임literally means alcohol game. search it on Youtube.