On this website: http://www.dwheeler.com/chess-openings/, it says that the Pirc Defense is
1. e4 d6 or
1. e4 g6. Which one of them is the true Pirc Defense?
1. e4 g6 could certainly transpose into a Pirc but it is not the first two standard moves. If you look at the ECO entry for B07, B08, B09 you will see the standard variations for the Pirc/Modern. I would say the link you give is wrong.
1. e4 d6
2. d4 without
2... Nf6 without
3. Nc3 without
3... g6 without
1. e4 d6 2. d4 Nf6 3. Nc3 g6 4. Nf3
1. e4 d6 2. d4 Nf6 3. Nc3 g6 4. f4
The Pirc and the Modern Defense are very closely related.
In the Pirc, black fianchettoes his king's bishop and plays ...Nf6; ...d6 is necessary to prepare ...Nf6, because otherwise white can chase the knight away with e4-e5. There are several setups white can choose, but black's defines the Pirc defence:
[fen "rnbqk2r/ppp1ppbp/3p1np1/8/3PP3/2N2N2/PPP2PPP/R1BQKB1R w KQkq - 2 5"]
His move order, 1...d6 and then 2...Nf6, makes sure that white's e4 pawn is attacked and that it can't advance; so white can't play a pawn to c4. That's important.
The Modern is like a Pirc without ...Nf6:
[fen "rnbqk1nr/ppp1ppbp/3p2p1/8/3PP3/2N2N2/PPP2PPP/R1BQKB1R b KQkq - 1 4"]
Now obviously, if black plays ...Nf6 here or within a few moves, then it's a Pirc again. But black has other moves as well: start something on the queenside with ...a6 or ...c6, develop the other knight to c6 or d7, or play ...e6 or ...e5 (and ...Ne7 later). But frequently black does end up playing ...Nf6 eventually and then it becomes a Pirc.
In this move order, white can also go for 3.c4:
[fen "rnbqk1nr/ppp1ppbp/3p2p1/8/2PPP3/8/PP3PPP/RNBQKBNR w KQkq - 0 4"]
Now again, if black doesn't play ...Nf6, then it's still a Modern Defence (one that also often arises after 1.d4 or 1.c4). But if black does play it, then it's not a Pirc, as in the Pirc white can't play c4. It's a King's Indian Defence instead! That normally starts 1.d4 Nf6 2.c4 g6 3.Nc3 Bg7 4.e4, but that's the same thing.
So -- fianchetto without Nf6 is the Modern, fianchetto with Nf6 but without a white pawn on c4 is the Pirc, fianchetto with Nf6 and c4 is the King's Indian.
There are also non-Pirc lines that start 1.e4 d6:
[fen "rnbqkb1r/pp2pppp/2pp1n2/8/3PP3/2N5/PPP2PPP/R1BQKBNR w KQkq - 0 4"]
In the Czech Defense (1.e4 d6 2.d4 Nf6 3.Nc3 c6), black doesn't play his bishop to g7.
[fen "rnbqkb1r/ppp2ppp/3p1n2/4p3/3PP3/2N5/PPP2PPP/R1BQKBNR w KQkq - 0 4"]
And this (1.e4 d6 2.d4 Nf6 3.Nc3 e5) is a way to get to the Philidor Defense, which normally starts 1.e4 e5 2.Nf3 d6 3.d4 Nf6 4.Nc3, to avoid several options by white, at the cost of allowing 4.dxe5 dxe5 5.Qxd8+, but that's thought to be OK for black.