My question is about the distinction between certain kinds of phase transitions. I understand what the difference between first and second order ones are. What is the difference between non equilibrium and equilibrium phase transitions?
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Probably all phase transitions are "out of equilibrium" in general, we simply approximate them as equilibrium phase transitions.
Equilibrium phase transition (using a Gibbs potential) usually means that both phases are in mechanical, thermal, and chemical equilibrium. So in other words, the pressure of both phases is equal, the temperature of both pases is equal, and the chemical potential of both phases is equal. We usually make the equilibrium approximation because the math is often more tractable, however in real life most phase transitions are not in equilibrium, yet often approximating them as equilibrium can yield important insights.
Usually a phase transition approximated as equilibrium is "slow" enough so that the change in pressure, temperature, and chemical potential is very small and can be approximated zero (e.g. these quantities are equal in both phases). However there are situations where the equilibrium approximation doesn't hold, for example in an explosion that may evaporate matter, the phase change appears violently and therefore it cannot be approximated in equilibrium.
