This is a very complex question.
First, by nature of moving first, white can clearly control whether the game is open, or not, more than black can, but black does have a say next. If white plays, 1. d4, 1. c4, 1. Nf3, or even moves like b3 or g3, we know that the game tends to be more closed than after 1.e4. After 1. d4, for example, black can attempt to play openings that are more open, like the Benoni or Benko. I do not know that I consider the King’s Indian more open as much as it is simply more complex, usually requiring operations from white to be on the queenside, while black plays on the kingside. That said, white has lines in response that are more positional than the Classical King’s Indian, which features all out attacks on opposite sides of the board.
I think that if you are white, it is relatively easy to steer the game to more closed, and positional, positions even in openings like the King’s Indian, by playing a variation with g3, for example. If you are black, it is clearly much harder to keep the position closed if white wants to play 1 e4, and open it up. There is a reason they categorize many openings with 1. e4 “open games”.
I think your strategies to keep things closed with black when white plays 1. d4 are relatively straight forward, especially since you can assume that white likes that type of position too, but what do you do against 1. e4? I think that you can play the French or Caro, but even something like the Kan Sicilian with 1 e4 c5 2 Nf3 e6 3 d4 cxd4 4 Nxd4 a6 keeps things more quiet, than playing a typical open Sicilian.
What do these openings all have in common? Of course, these are not all exactly the same for every position, but they all tend to have either less pawn breaks. You almost always have to allow one pawn break, or clash of pawns, but if you do not acquiesce to multiple pawn breaks, there is less likelihood that the position will open a lot. Pawn breaks usually lead to pawn trades, which leads to a more open position, so try to avoid pawn clashes if you are trying to keep the position closed. That said, they are not unavoidable, and you have to judge whether you need to fight back for space. If you sit back forever, you will be smothered to death eventually. The goal is like the Hedgehog where you sit back, and develop, and only then, start to fight back with your pawn break.
Although I have never thought about your question directly, I think we all make these choices subconsciously when we are developing our opening repertoire. I certainly play 1. d4, and usually g3 to get the Catalan lately, knowing that the game will be more closed and positional. It is funny that relatively few of my opponents play the open Catalan with cd, and prefer c6 as a rule…that suits me fine.
Again, this is very complex, and there are no hard-and-fast rules, so I may have missed something, but hopefully, I gave you some food for thought.