The layered FeTe2O5Cl compound was studied by specific-heat, muon-spin relaxation, nuclear magnetic resonance, and dielectric as well as neutron and synchrotron x-ray diffraction measurements, and the results were compared to isostructural FeTe2O5Br. We find that the low-temperature ordered state, similarly as in FeTe2O5Br, is multiferroic: the elliptical amplitude-modulated magnetic cycloid and the electric polarization simultaneously develop below 11 K. However, compared to FeTe2O5Br, the magnetic elliptical envelope rotates by 75(4). and the orientation of the electric polarization is much more sensitive to the applied electric field. We propose that the observed differences between the two isostructural compounds arise from geometric frustration, which enhances the effects of otherwise subtle Fe3+ (S = 5/2) magnetic anisotropies. Finally, x-ray diffraction results imply that, on the microscopic scale, the magnetoelectric coupling is driven by shifts of the O-1 atoms, as a response to the polarization of the Te4+ lone-pair electrons involved in the Fe-O-Te-O-Fe exchange bridges.