The nonlocal response of plasmonic materials and nanostructures is often described within a hydrodynamic approach, which is based on the Euler-Drude equation. In this paper, we reconsider this approach within an extension proposed by Halevi [Phys. Rev. B 51, 7497 (1995)]. After discussing the impact of this extended model on the propagation of longitudinal volume modes, we reevaluate within this framework the Mie scattering coefficients for a cylinder and the corresponding plasmon-polariton resonances. Our analysis reveals a nonlocal, collisional, and size-dependent damping term, which influences the resonances in the extinction spectrum. A transfer of the Halevi model into the time domain allows to identify a contribution to the current, which shares similarities with Cattaneo-kind diffusive-wavelike dynamics. After a comparison to other approaches commonly used in the literature, we implement the Halevi model into the discontinuous-Galerkin time-domain finite-element Maxwell solver and identify an oscillatory contribution to the current. Such an implementation of the Halevi model in time domain is of particular importance for applications in nanoplasmonics where nanogap structures and other nanoscale features have to be modeled efficiently and accurately.