Finite volume (FV) methods for solving the two-dimensional (2D) nonlinear hyperbolic conservation laws on unstructured meshes are well known and applied for some time now. There are mainly two basic formulations of the FV method: node-centered (NCFV) and cell-centered (CCFV). For both formulations, details will be given of the development and application of a second-order well-balanced Godunov-type scheme. Using a controlled environment for a fair comparison, a complete assessment of both FV formulations is attempted through rigorous individual and relative performance comparisons to the approximation of analytical benchmark solutions, as well as to experimental and field data. To this end, an extensive evaluation is performed using different time dependent and steady-state test cases for the nonlinear shallow water equations and the Euler equations. These test cases are chosen as to compare the performance and robustness of each formulation under certain conditions and evaluate the effectiveness of a novel procedure for multidimensional solution reconstruction and edge-based limiting for the CCFV approach. Emphasis to grid convergence studies is given, with the grids used to range from regular grids to irregular ones with random perturbations of nodes.