In positron emission, a proton decays into a neutron, electron, and neutrino. Since the mass of a proton is less than that of a neutron, does that mean that energy is converted into mass in the reaction? Beta decay#β+ decay says that some of the binding energy goes into converting a proton into a neutron, but that's not really what I'm looking for. So, does the extra mass result from the conversion of energy into mass?
The distinction between energy and mass is really a matter of semantics here and is not really a fundamental distinction. We tend to call it "mass" when energy is in a bound state, i.e. an object such as a proton or an atomic nucleus that more or less holds together and stays in one place.
You can call positron emission "energy to mass conversion" if you like, but that's really an arbitrary distinction between the supposed "mass" of a proton/neutron and the "energy" of a nuclei. They're both energies of various kinds when you look close enough anyway!
The mass of a proton is mostly not the mass of the three valence quarks, but the energy in the quark-gluon field that surrounds them. Similarly, the mass of an atomic nucleus is not just the total mass of all the protons and neutrons in it; there is a significant correction to the mass from nuclear binding energy (which contributes negatively, reducing the total mass).
Even the masses of quarks, electrons, neutrinos etc. is really an energy of interaction with the Higgs field.
So most-to-all of what you ordinarily call "mass" is actually a sum of various energies of constituents and binding energies for different levels of structure.