rshalom: Rav Neiman shlit”a is 100% correct. There is no so-called “apparent discrepancy” c”v with Rav Moshe (as you put it) whatsoever. Rav Moshe never ever said its okay to shake a woman’s hand. Never. To no one. Rav Moshe has 3 psaks in Igros Moshe yelling assur assur assur in strong language. Three teshuvos, to make it exceptionally clear because he apparently realized some folks (as we see here) will come crying he allegedly said otherwise.
Igros Moshe EH I #56 page 144;
Igros Moshe OC I 113 page 177;
Igros Moshe EH IV 32.9 page 76.
[p.p. 130-131])