Yes, the way you interpreted the motion is correct, and it is more downward than forward.
Do whatever is comfortable for you. My bathroom is in the basement with just a pedestal sink so I don't have any good place to set razors down. For that reason I continue to hold the razor in my right hand and rinse off my face and re-lather with my left. I always figure the razor is safer in my hand in my bathroom and at the gym. If you have a nice wide counter-top surface, you can put the razor down on a wash cloth, or something else soft, to protect the razor.