Like what...? The second scenario you presented isn't exclusive to women though; Westerners include males as well. I can totally understand if a person refused to shake my hand because they are paranoid about cleanliness or something (like that Howe guy from "Deal or No Deal" with the fear of germs) and I would respect their decision since they'd also treat everyone else like that (it's a consistent rule).
Someone refusing to shake my hand *solely* because I happened to be born with a vagina, however, is plain ridiculous imo. What is the logical justification of that rule? Shaking hands is more of a professional thing, it's not hugging or even something personal/s-xual...
Apparently it leads to sin.