How is a phase equilibrium defined for a one-component system?
Melting is a process. A process is not a state. Since only states can be &quot;at equilibrium&quot;, it doesn't make sense to speak of a real melting substance as being in an equilibrium state. However, melting is deemed reversible if performed infinitely slowly, and during such a transition the system passes through a series of equilibrium (or &quot;quasi-equilibrium&quot;) states. But such a process is an idealization.  In reality, you can't &quot;melt ice isothermally&quot;. Real transitions are therefore irreversible because system and surroundings are not actually in thermal equilibrium during heat transfer$^\dagger$.
Note that if you place in contact two of the intermediate states sampled during the transition neither will change, that is, each remains in its original equilibrium state.
The second definition encompasses the first one (equality of chemical potential across phases). The second definition therefore better captures what is meant by phase equilibrium. The conditions for phase equilibrium are those used to derive the phase rule, namely mechanical, chemical and thermal balance between phases. A phase is a region of uniform composition, temperature and pressure.

$^\dagger$ We say that it is possible to melt a pure substance using an ideal reversible process that leads the system infinitely slowly through a series of equilibrium states (each intermediate stage in that reversible process is an equilibrium state).  In order to move between intermediate states, an infinitesimally small amount of heat has to be transferred from the infinitesimally warmer surroundings (exploiting, say, something like a Maxwell demon). In what is a logical sleight of hand, we say that the temperature difference is so small however, that the system and surroundings are effectively isothermal. This does not reflect what happens in a real process.