The description of electron-phonon coupling in materials is complex, with varying definitions of coupling constants in the literature and different theoretical approaches available. This article analyzes different levels of theory to introduce and compute these coupling constants. Within the quasi-particle picture, we derive an effective linear-coupling Hamiltonian, describing the interaction of electronic quasi-particles with vibrations. This description allows a comparison between coupling constants computed using density functional theory and higher-level quasi-particle approaches by identifying the Kohn-Sham potential as an approximation to the frequency-independent part of the self-energy. We also investigate their dependence on the exchange-correlation (XC) functional. Despite significant deviations of the Kohn-Sham eigenvalues, which arise from different XC functionals, the resulting coupling constants are remarkably similar. A comparison to quasi-particle methods, such as the well-established G0W0 approach, reveals significant quasi-particle weight renormalization. Surprisingly, however, in nearly all the considered cases, the coupling constants computed in the DFT framework are excellent approximates of the ones in the quasi-particle framework, which is traced back to a significant cancellation of competing terms. Other quasi-particle methods, such as the Outer Valence Green's Function approach and the Δ