Does draft or best preview quality have anything to do with the final output quality or is draft just to cut down on render time?

Under file --> render as is where you set render quality.

The thing under video preview is just for preview.

Where do you see render as under File? I am working with 6.5 if that makes a difference. I'm talking about the settings in the effects control box. There is a small arrow at the top right that creates another menu and these draft and best options are at the bottom of that menu.