In this talk, we present the three-loop QCD conversion factor from the regularization-independent massive symmetric-momentum-subtraction (mSMOM) mass to the $\overline{\mathrm{MS}}$ mass $\overline{m}$, extending the previous result by two perturbative orders. (Interesting revisions of the mSMOM conditions are discussed too.) A preliminary application to the RBC/UKQCD 2024 analysis indicates that its central charm-quark mass would decrease by about one percent and move toward the earlier FLAG averages. We also discuss an alternative matching route based on the trace-anomaly-subtracted heavy-quark mass $m_{\sigma}$, exploiting its nice properties including scale/scheme-independence (and hence a unique value for each heavy-quark flavor) and the absence of the leading infrared-renormalon ambiguity. We present a compact conversion formula relating it to the perturbative pole mass through five loops and to the $\overline{m}$ through four loops. Furthermore, we outline how $m_{\sigma}$ for a heavy quark can in principle be determined from lattice QCD through an intermediate regularization-independent renormalization and perturbative matching, for which we analyze the matching factor through four loops with appealing features revealed. These properties motivate $m_{\sigma}$ as a possible common reference definition for comparing heavy-quark masses obtained in different schemes.