I hold my razor mainly like your photo on the right. I use a similar grip to your photo on the left when going ATG on my neck.
I don't really think about it, somehow my hand finds the optimal position for balance and effect when holding a razor I'm not accustomed to.
My answer is a resounding YES! I change my grip on the razor depending on which pass I am on.
Luc:This could help: http://wiki.badgerandblade.com/Safety_Razor_Grips