Using 24.5 million psi(2S) decays collected with the CLEO-c detector at CESR we present the most precise measurements of magnetic dipole transitions in the charmonium system. We measure B(psi(2S)->gamma eta_c) = (4.32+/-0.16+/-0.60)x10^-3, B(J/psi->gamma eta_c)/B(psi(2S)->gamma eta_c) = 4.59+/-0.23+/-0.64, and B(J/psi->gamma eta_c) = (1.98+/-0.09+/-0.30)%. We observe a distortion in the eta_c line shape due to the photon-energy dependence of the magnetic dipole transition rate. We find that measurements of the eta_c mass are sensitive to the line shape, suggesting an explanation for the discrepancy between measurements of the eta_c mass in radiative transitions and other production mechanisms.