Why are diffusion equations typically formulated using $-\partial_{x}(D\partial_{x}u)$ when Fokker--Planck has $-\partial_{xx}^{2}(Du)$

Question

Is there any physical process that naturally gives rise to equations of the form $$ \partial_{t}u -\partial_{x}(D\partial_{x}u) $$ or is this form really used in all my textbooks (e.g. Evans) and lecture notes simply because it lends itself more easily to partial integration than the Fokker-Planck form $$ \partial_{t}u - \partial_{xx}^{2}(Du)\quad ? $$ I have found the Wikipedia article on anisotropic diffusion using the first form, but that just adds to my confusion because I would have thought that anisotropic diffusion, as any diffusion, would surely have to be in Fokker-Planck form. To cap it all, trying to get an intuitive idea of the two equations, I noticed that my understanding of diffusion actually matches more with the first form rather than the Fokker-Planck equation that is supposed to describe the resulting densities: effective flow equals gradient times conductivity factor, resulting changes equal divergence of flow.