No idea what kind of explanation you are looking for. First derivation uses well defined assumptions - if they are wrong, ratio is probably far from 1.

Your example (0.001M HA^{-}, pKa1,2 = 3 & 7) yields a solution with pH 5.15 (instead of expected 5) and ratio

[tex]\frac{[H_2A]}{[A^{2-}]} = \frac{6.9\times10^{-6}}{1.4\times10^{-5}}=0.49[/tex]

Apparently you can't use approximate formula for this case. Hardly surprising, tables below shows that below 0.01M one has to expect problems.