First-principles calculations of phonons are often based on the adiabatic approximation and on Brillouinzone samplings that might not always be sufficient to capture the subtleties of Kohn anomalies. These shortcomings can be addressed through corrections to the phonon self-energy arising from the low-energy electrons. The exact self-energy involves a product of a bare and a screened electron-phonon vertex which have been proposed as a reliable approximation for self-energy differences [Phys. Rev. B 82, 165111 (2010)]. We assess the accuracy of both approaches in estimating the phonon spectral functions of model Hamiltonians and the adiabatic low-temperature phonon dispersions of monolayer TaS2 and doped MoS2. We find that the approximate method yields excellent corrections at low computational cost, due to its designed error cancellation to first order, while using a bare vertex could in principle improve these results but is challenging in practice. We offer an alternative strategy based on downfolding to partially screened phonons and interactions [Phys. Rev. B 92, 245108 (2015)]. This is a natural scheme to include electronelectron interactions and tackle phonons in strongly correlated materials and the frequency dependence of the electron-phonon vertex.