Both, actually. The vertical motion of the storm lifts the sand vertically, and the winds advect the sand laterally.
Yes, the sand is of course lifted off the ground, but the amount would not be visible in this photo. Even if it were lifted up into the air as high as the top of Mount Everest, the vertical component would only be equal to five pixels of this image (less than 1% of its width).
It's easy to overestimate the "thickness" of the Earth's atmosphere, or geographic "roughness". If the Earth were the size of a billiard ball, it would be about as smooth *as* a billiard ball, even taking into account such "imperfections" as the depth of the Marianas Trench and the height of Everest. At billiard-ball size, the difference in heights of those features would amount to imperfections no greater than a few one-thousandths of an inch.
I don't have enough background in remote sensing to assess the pixel by pixel attributes of the vertical mixing versus horizontal diffusion and/or advection of the sand in the image.
Just wanted to clarify that the physical process responsible was doing both.
By the way, your analogy of the thickness of the atmosphere (and significant orographic features) comes as a new way of explaining it to me and I like it. I was used to the old standby of a dime on a common globe, but hadn't heard it from a pool ball perspective...