Consider the following three “levels of fairness”, starting with the “weakest” (each stronger level of fairness actually implies the weaker levels):
To the casual reader “place-fairness” and “permutation-fairness” might sound like the same thing, but they are different. It is quite possible for a set of dice to be “place-fair” without being “permutation-fair”. In such a case, each player would be equally likely to end up in any position of the hierarchy, but some overall orderings of players would be more likely to occur (for example, if the dice are colored Black, Red, Green, and Blue, any color would be equally likely to end up in any of the four sorted positions, but maybe “Green, Red, Black, Blue” ordering comes up more often, than say, “Red, Blue, Black, Green” – individually the colors are equally likely to end up in positions 1-4, but how their positions relate to the other colors might not be equi-probable). Achieving that equi-probability of the permutations only happens with “permutation-fairness”.
The original problem of looking for a go-first-fair set soon morphed into trying to find a set that was permutation-fair. So, calling this wiki the “Go First Dice Wiki” is actually a misnomer; it should be called “The Permutation-Fair Dice Wiki”, but that does not have as nice a ring to it, and the original phrasing has stuck.