For White, which variation of the King's Indian Defence tends to lead to the most solid, positional, slow, strategic, quiet and closed positions? I am a 1750 Elo player with a very positional, quiet and strategic style of play.

I would like to know which variation I should play when, as White, I face the King's Indian Defence: 1...Nf6 2...g6

The variation must meet these two conditions:


On average it must tend to lead to positions which are as much solid, positional, slow, closed, quiet and strategic as possible (I want nothing sharp and tactical).
I should be able to play it both by beginning with 1. d4 Nf6 2. c4 and by beginning with 1. Nf3 Nf6 2. c4


Don't hesitate to suggest me more than one variation, so that if I don't like the first variation maybe I shall like the second one.
According to Joe Gallagher-Starting Out The King's Indian Defense and David Vigorito-Attacking Chess-The King's Indian Volume 2 the most solid line for White in King's Indian is the Fianchetto variation. I tend to agree with it. 

Before explaining why, let me just show you the tabya so I can better explain my point of view :

[Title "KID, the Fianchetto variation"]
[fen ""]

1.d4 Nf6 2.c4 g6 3.Nf3 Bg7 4.g3 O-O 5.Bg2 d6 6.O-O Nc6


After briefly looking through the lines ( there is a vast amount of work to review ) I came to the same conclusion as the above writers based on the following :


Center remains fluid, so Black can not pursue standard ...f5 idea as in other lines;
Because of the first point, and the fact that fianchetto castling is very hard to crack, White king is ultra safe ;
White has more space;


It is not easy for Black to find counterplay, but he can reach equality if he knows how to play this line. Tactical shots are limited to the rook sacrifices along the b-file so Nf6 + Bg7 battery can profit from discovered attack, but otherwise things are positional and dynamic. 

There are several sub-variations here, but I would recommend you to play Panno line, since Yugoslav attack is quite sharp and theoretical.

[Title "KID, the Fianchetto variation-Panno line"]
[fen ""]

1.d4 Nf6 2.c4 g6 3.Nf3 Bg7 4.g3 O-O 5.Bg2 d6 6.O-O Nc6 7.Nc3 a6 8.h3 Rb8 9.e4 b5


White has 2 options here, cxb5 and e5 but I advise you cxb5 as it is safer one. Play will become dynamic, but you have dodged a lot of theory and sharp tactical mayhem from Classical line or Saemish attack... For detailed coverage of this line get Vigorito's book. It is printed in 2011 and is very well written, in my opinion.

Do not forget that KID is a sharp opening and that you can not "sterilize it", you can only reduce the level of sharpness and that is it.