The interfacial Dzyaloshinskii-Moriya interaction (iDMI) is of great interest in thin-film magnetism because of its ability to stabilize chiral spin textures. It can be quantified by investigating the frequency nonreciprocity of oppositely propagating spin waves. However, as the iDMI is an interface interaction, the relative effect reduces when the films become thicker, making quantification more difficult. Here, we utilize all-electrical propagating spin-wave spectroscopy to disentangle multiple contributions to spin wave frequency nonreciprocity to determine the iDMI. This is done by investigating nonreciprocities across a wide range of magnetic layer thicknesses (from 4 to 26 nm) in Pt/Co/Ir, Pt/Co/Pt, and Ir/Co/Pt stacks. We find the expected sign change in the iDMI when inverting the stack order and a negligible iDMI for the symmetric Pt/Co/Pt. We additionally extract a difference in surface anisotropies and find a large contribution due to the formation of different crystalline phases of the Co, which is corroborated using nuclear magnetic resonance and high-resolution transmission-electron-microscopy measurements. These insights will open up avenues to investigate, quantify, and disentangle the fundamental mechanisms governing the iDMI, and pave a way toward engineered large spin-wave nonreciprocities for magnonic applications.