I am unclear about the adjective "first" or "second" used in phase transitions. Take the liquid-gas transition for example. If we vary the volume of the system at constant T, at some point we will have two phases co-existing. The relevant free energy of the complete system (liquid+gas), the Helmholtz free energy $A$, varies continuously such that its second derivative is zero and first derivative is constant. On the other hand between the phases themselves $A$ is discontinuous, but intensities like T, P etc are same. If one studies the variation along a P-T isotherm (vary pressure of the system at constant temperature), the relevant free energy of the total system $G$ is continuous but non-differentiable at the transition. Exactly what is the criterion for calling this "first" order?
I have checked wikipedia and a few online lecture notes, and which free energy derivatives should one consider is not clear to me.
EDIT: I just realized a similar question has been raised a few days ago. Not sure how it slipped my attention. However, in light of that one, my question can be simplified to which free energy do I choose for deciding order? I can study a phase transition using different thermodynamic potentials, the free energy being the one that corresponds to my independent variables.