Solving partial differential equations (PDEs) is one of the most traditional tasks in scientific computing. In this work, we consider numerical solutions of initial value problems (IVPs) problems partly or entirely given by linear PDEs and how to compute solutions with a method we refer to as rational approximation of exponential integration (REXI). REXI replaces a typically sequential timestepping method with a sum of rational terms, leading to the possibility to parallelize over this sum. Hence, this method can potentially exploit additional degrees of parallelization for scaling problems limited in their spatial scalability to large-scale supercomputers. The main contribution of this work lies in developing the "unified REXI" in which we show algebraic equivalence to other methods developed up to five decades ago. Such methods cover, e.g., diagonalization of the Butcher table for implicit Runge-Kutta methods, Cauchy-contour integrationbased methods, and direct approximations. To our best knowledge, this is the first time of such a comparison and deep investigation of all these methods. Finally, we will show the applicability of REXI to the nonlinear shallow-water equations on the rotating sphere, including HPC results. While previous REXI studies have focused on exposing more parallelism to enable faster time to solution, we also consider efficiency at prescribed accuracy and find that diagonalized Gauss Runge-Kutta methods (formulated as REXI) are compelling highly efficient methods.