Direct detection experiments can probe relatively low-mass dark matter (DM) by measuring electron recoil signals. In this talk, I will present our recently developed theoretical framework for analyzing the dark matter-bound electron scattering within the effective field theory frameworks. For the most general DM-electron nonrelativistic or relativistic interactions for DM with spin up to 1, the event rate in this formalism decomposes into a sum of three terms, each factorized into a DM response function and a generalized atomic response function, with no cross-dependence between DM properties and atomic structure. I will illustrate how experimental data can be mapped onto constraints for effective interactions. As a concrete application, I will present corrected constraints on the fermionic DM anapole moment, addressing an error previously overlooked by both theoretical and experimental studies.