If I’m understanding correctly, I’ve always played the second version.
For what it’s worth, games writer David Parlett says the second version is correct (take the upcard or pass first round). He’s written on the history of Gin (and that’s what he says is the correct name, not Gin Rummy) somewhat extensively so I think his word carries some weight.
So not only may you have been playing the wrong game, you might have been calling it the wrong thing too!
Yeah, and the upcard deadwood variation mentioned above (up card determines the maximum amount of deadwood you can have to knock, with an ace requiring gin) we called “Oklahoma Gin.” And settling the money (or the score) after a match was penny a point, quarter a box, dollar a game (game being to 100), if I remember right.