Well, that is a way to define "fair", but my initial comment was that *that* doesn't hold for the first method presented since each of the possible permutations has the same parity.
(The problem of a "random" shuffle is a real one and not word play at all. Consider a dealer shuffling cards in a poker game; how should he do it to maximize the randomness of the result - should he shuffle three times, 7 times, ...? Should the shuffles be somewhat careless (so the cards son't interleave perfectly? (Yes to latter!) Here the idea is that there will be some fixed shuffling routine (e.g. shuffle, shuffle, box, shufffle), but one wants the result to be sufficiently randomized. This isn't precisely the same thing as what the OP asked, but is often what is actually desired when one discusses the question of a "fair" or "random" shuffle. This seemed to be alluded to in one of Zaxo's replies, and as I mentioned it has been analyzed in the liturature.)
chas