Previous nonlinear spinning disk models neglected the in-plane inertia of the disk since this permits the use of a stress function. This paper aims to consider the effect of including the in-plane inertia of the disk on the resulting nonlinear dynamics. The inclusion of the in-plane inertia results in coupling between the in-plane and transverse vibrations of the spinning disk. The full nonlinear equations are simplified to a simpler nonlinear two degrees-of-freedom model via the method of Galerkin. This 2 DOF model is analyzed with a canonical perturbation approach and with numerical simulations. Through the use of these analytical and numerical tools, it becomes apparent that the inclusion of in-plane inertia gives rise to new phenomena in the dynamics of the system that are not predicted if the in-plane inertia is ignored.