Using 2.45×107 ψ(2S) decays collected with the CLEO-c detector at the Cornell Electron Storage Ring we present the most precise measurements of magnetic dipole transitions in the charmonium system. We measure B(ψ(2S)→γηc)=(4.32±0.16±0.60)×10-3, B(J/ψ→γηc)/B(ψ(2S)→γηc)=4.59±0. 23±0.64, and B(J/ψ→γηc)=(1.98±0.09±0.30) %. We observe a distortion in the ηc line shape due to the photon-energy dependence of the magnetic dipole transition rate. We find that measurements of the ηc mass are sensitive to the line shape, suggesting an explanation for the discrepancy between measurements of the ηc mass in radiative transitions and other production mechanisms.