Here in the United States, a firm but gentle grip is the preferred style, for both men and women. If you offer your hand drooping down I feel like you believe you are lowering yourself by being forced to shake hands with me.
If you try to break the bones in my hand, I get the feeling you are trying, again, to prove some sort of superiority.
Why people use these awful handshake styles is beyond me. I would not shake hands if I knew in advance that the other person was going to give the limp banana peel or the vise grip.