Energy eigenstates are states where x and t can be separated in the time-dependent equation, so the solutions can be factored into two terms, as f(x)exp(iwt). Solutions with no potential-energy function (V(x,t) = 0) are traveling plane waves, and solutions with an infinite potential well are not dispersive. So calling the Schrödinger equation a "diffusion equation" seems misleading to me, and "wave equation" seems reasonable. I think the linear time derivative in the Schrödinger equation is misleading because its coefficient is imaginary. I think it has more in common with a real-valued second derivative from a physical or dynamical point of view, even though it superficially looks like a diffusion term. Traveling waves aren't defined in terms of oscillation, but standing waves are certainly associated with it, and classical diffusion is a completely different phenomenon, with no oscillation at all.
PS: How do you edit equations here?