Not to stifle further discussion, but this Wikipedia page has a wealth of examples

https://en.m.wikipedia.org/wiki/List_of_placeholder_names_by_language

Rather than trying to define abstractions for turns and rounds, maybe you could implement the game state (and possibly individual players) as a finite state machine.

In this model, the way you set up the states and transitions for a particular game will capture its idiosyncracies in turn and round order.

Sounds similar to how the monarch of Malaysia is chosen. https://en.m.wikipedia.org/wiki/Succession_to_the_Malaysian_thrones

The next national ruler is chosen from the nine hereditary state sultans. It's technically an elective monarchy, but there is a conventional rotation order among the states.