The
chessgames.com link you provided classifies the Pirc as specifically 1.e4 d6 2.d4 Nf6, not "everything that opens 1.e4 d6".
After 1.e4 d6, it correctly shows that any of the following could occur:
Pirc
Philidor
Robatsch
Uncommon
King's Indian Attack
Modern
Detecting unusual move orders and transpositions is not the most straightforward thing to do.
In some situations, you would want the opening to be classified by the exact order of moves played, and not by whatever position it happened to transpose to.
For example, the exact same position arises from both of the following:
1.e4 e6 2.d4 d5 3.exd5 exd5 4.c4 dxc4 5.Bxc4
1.d4 d5 2.c4 dxc4 3.e3 e5 4.Bxc4 exd4 5.exd4
The first is quite clearly a French Exchange, and the second is quite clearly a Queen's Gambit Accepted.
However, that position is reached 5 times as often from the QGA move order, so if I reach that position from the French Exchange should it get classified as a QGA?
In that case, using the exact move order to classify the opening makes the most sense.
In other cases, using the transposed position might make more sense, but it's really hard to determine when one is appropriate and the other isn't.
I tend to prefer the exact move order approach, because if a common position is transposed to by an unusual move order, that usually (not always, of course) means that there were probably other approaches to playing the position unique to that exact move order, so I wouldn't want a game that happened to transpose to a Pirc to be classed that way, because it's different from "normal" Pircs in that there could be unique options based on the move order.
To use an extreme example, imagine a game that starts 1.e4 Nf6 2.d4?? d6??. Quite aside from the fact that it started as an Alekhine and is now (temporarily, perhaps) a Pirc, classifying it as a Pirc makes it seem that the position was arrived at sensibly, and glosses over the fact that it's actually a terribly misplayed Alekhine, where white dropped his pawn and black didn't exploit that.
To continue that example, the other problem comes from move orders that transpose through a couple openings.
For example, if that game continues 1.e4 Nf6 2.d4?? d6?? 3.f3 g6, a sideline of the Pirc, and continues 4.c4 Bg7 5.Nc3, should that game stay classified as an Alekhine since it started that way, a Pirc since it reached a Pirc position, or get changed to a KID, Samisch variation, since the position now corresponds to that opening?
Usually sites/programs that recognize transpositions answer that question by classifying it as whatever opening it last corresponded to, which is fine, but somewhat arbitrary, and doesn't help in cases like my first example, where it's actually the exact same position from two very different openings.
In general, I think trying to recognize transpositions is too difficult and arbitrary to be worth the effort. If you arrive at a common position from an uncommon move order, some uncommon classification makes sense. Using the exact move order is the only non-arbitrary (and non-misleading) way to uniquely classify what was played.
It also makes more sense from a study perspective. If you're preparing for an opponent or trying to improve your repertoire, there's a big difference between playing the same opening a lot by the same move order, and playing the same opening a lot by various move orders and transpositions
The latter requires different preparation, since again, different move orders usually mean different options you have to be aware of (so much so, in fact, that Soltis wrote a whole book about tricking people that way,
http://www.amazon.com/Transpo-Tricks-Chess-Batsford-Books/dp/0713490519, although to be fair, Soltis will write a book about anything).
Just my $0.02. Both approaches have their appeal, and neither is "wrong" (although as you can probably tell I think a solid case can be made for using the exact move order approach over the transpositional). Lichess has picked one approach, and that's fine by me :)