Phase lag changes cross spectrum
Derivation seems to be wrong. I am guessing from derivation that it is power spectral density that you are talking about. Your derivation seems to be wrong, cross density would be $$S_{xy}(x)S_in=G1(x)G2\times(G2^*)(x)S_in$$ or $$S_{yx}(x)S_in=G1\times(G1^*)(x)G2(x)S_in.$$ If you take magnitude spectrum both will be same, phase spectrum will be of 180 shift.