Yes, it probably has come up somewhere, but it is not very well known. I also tried the type of chess where you win if you give a check, but white wins easily as I shall show.
Different types of chess
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Edit:5.Qd4! with both f6 and Nf6 wins much more quickly (after f6 I accidentally put Qe4)
There are many different types of chess, like Greek chess (where capturing is prohibited) and Scotch chess(white has one move, black has two, white has three etc.). But I invented my own type of game: the player who gives a check without the piece being able to be taken is the winner, and if a check is given, the piece has to be taken(if possible). Try it out!