The search for sub-GeV dark matter (DM) particles via electronic transitions in underground detectors attracted much theoretical and experimental interest in the past few years. A still open question in this field is whether experimental results can in general be interpreted in a framework where the response of detector materials to an external DM probe is described by a single ionisation or crystal form factor, as expected for the so-called dark photon model. Here, ionisation and crystal form factors are examples of material response functions: interaction-specific integrals of the initial and final state electron wave functions. In this work, we address this question through a systematic classification of the material response functions induced by a wide range of models for spin-0, spin-1/2 and spin-1 DM. We find several examples for which an accurate description of the electronic transition rate at DM direct detection experiments requires material response functions that go beyond those expected for the dark photon model. This concretely illustrates the limitations of a framework that is entirely based on the standard ionisation and crystal form factors, and points towards the need for the general response-function-based formalism we pushed forward recently. For the models that require non-standard atomic and crystal response functions, we use the response functions of to calculate the DM-induced electronic transition rate in atomic and crystal detectors, and to present 90% confidence level exclusion limits on the strength of the DM-electron interaction from the null results reported by XENON10, XENON1T, EDELWEISS and SENSEI.