KID is defined by 1. d4 Nf6 2. c4 g6 with pawn structure 2CD1g.
King's Indian Defence - Black kingside attack
Apologies, post #1 shows the finishing position of the King's Indian Defence (as defined by Burgess). The actual moves are:
So is Burgess wrong? Is Yigor right? Please base your answer on a *serious* publication like Paul van der Sterren's "Fundamental Chess Openings", not Wikipedia or some database app...

In the Encyclopedia of Chess Openings KID has ECO codes E60-E99. The basic KID (E60) is defined by 1. d4 Nf6 2. c4 g6.
In the Encyclopedia of Chess Openings KID has ECO codes C60-C99. The basic KID (C60) is defined by 1. d4 Nf6 2. c4 g6.
You, and some online searching, have convinced me. Would you agree that what Burgess gives us is the main line of the King's Indian Defence? Even some of Burgess' example games have different third moves, though. So he has no excuse.
Burgess has a separate entry on the Grünfeld Defence. I think he's tried to simplify the classification for his beginners book and it hasn't quite worked. In any case, I think his book is still useful, for beginners.
He suggests there are five main "systems" of the King's Indian:
The Classical, which follows the scenario outlined in post #1:
The Samisch, he describes as a "subtle blockading system":
The Averbakh he declares "flexible" for White.
The Four Pawns Attack is White's most overtly aggressive system:
Finally there are system where white avoid e4, but that's enough to consider for now! Which of these systems do you prefer, and why? Or do you prefer something else?

Mal_Smith: Yes, 3. Nc3 Bg7 is the main line (E62-E99). Chess.com Explorer calls it KID: Normal Variation.

1. d4 Nf6 2. c4 --- At this point the game is classified as some form of "A" Queen's Pawn
It becomes a KID with 2. ... g6 [ I thought this was E60 ]
3. Nc3 ...
It is at this point we have a major branch for black along the lines of 3. ... d5 [ D80 Gruenfeld Defense ] or 3. ... Bg7 [ E61 KID ]
It is move 4. typically 4. e4 that ends a Gruenfeld cross over, the point is 4. Nf3 perfectly playable can transpose into unfavourable Gruenfeld Exchange Variations.
Just seen Micky has posted this with #4
Grünfeld Gruenfeld is like Defence or Defense
1. d4 Nf6 2. c4 g6 3. Nc3 Bg7 4. e4 d6 5. Nf3 O-O 6. Be2 e5 7. O-O Classical E92
1. d4 Nf6 2. c4 g6 3. Nc3 Bg7 4. e4 d6 5. f3 O-O 6. Be3 e5 7. d5 Saemisch E87

The King's Indian Defence, according to Burgess, is defined by the moves . . . black will try to create a pawn chain from c7 to g3, attacking with pieces on the kingside behind this advanced pawn chain. White builds a parallel pawn chain and attacks on the queenside . . . What do people think about this strategy for Black? Is there a better strategy for Black? Is there a better response to 1. d4!?
To me that's the quintessential KID position. Black goes for an all out mating attack, often sacrificing multiple pieces to break through. White wins some queenside pawns, and if the attack is survived, then can win the endgame.
It's a perfectly sound strategy, but I personally dislike this type of game because first of all I dislike sacrificing when I can't calculate to mate, and without a feel for when an attack will work, sometimes these attacks seem mysterious to me. It's actually something I'm currently working on learning, so I might switch to this defense heh.
That said, white is not forced to play d5. There are many opportunities to instead play dxe. In your diagram as early as move 7. Also you post some other white setups yourself, I think the Samisch in particular is popular for club players. So even though I'd like more experience with these attacks, the KID probably isn't the ideal opening for me to pick.
One of my early attempts at the KID met with this sort of idea by white:
When the f5 break can actually backfire. Notice white's king isn't castled yet, and can even be safe in the center for a long time. This made the game strategically too complex for me (I tried f5 anyway and lost).
Long story short, it's a complex defense, and not as simple as "get that pawn chain and go all out on the kingside every game."
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Your last question is if there's a better response to 1.d4. I think the QGD is great. The ideas are classical (i.e. makes sense to sub-masters) and it serves as a good foundation to later adopt more complex defenses like the Nimzo and QID (which make a good pair). I group the KID with the Grunfeld as openings for players who have both learned and moved beyond the classical ideas, and also players who are willing to invest a lot of time learning opening theory.
But of course the QGD isn't a training wheels opening either. It's been used by every world champion in top level play. It has one of the best (maybe even the best) pedigrees in all of chess.
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In the end pick something you enjoy playing. Even in highly difficult openings, if you're confused, your similarly rated opponents will likely be just as confused. I don't want to say the KID is a bad choice, but for the reasons above I don't think it's a good choice for me at least.

Oh, and in case you think that opening (with g4) was just some trash I happened to see from a club player, here's the highest rated game in my database featuring that idea:
And to be fair. One where black wins.
Funnily enough, we see the opposite. White going for the kingside and black the queenside.
"This, to me, is real chess. The King's Indian is a cross between all out warfare and a fairyland where incredible sacrifices and sensational brilliances are possible" - Graham Burgess, The Mammoth Book of Chess. Black allows White to build up a big centre, and then "more normally" attacks on the kingside.
The King's Indian Defence, according to Burgess, is defined by the moves:
In the kingside attack black plays e5, and "somehow" induces White to play the e5 advance. Then black will try to create a pawn chain from c7 to g3, attacking with pieces on the kingside behind this advanced pawn chain. White builds a parallel pawn chain and attacks on the queenside, turning right to attack to prevent reinforcements mating the white King. Here's a game illustrating this:
What do people think about this strategy for Black? Is there a better strategy for Black? Is there a better response to 1. d4!?