thermodynamics - Why the non-analyticity of free energy function implies phase transition? And what's its connection with other 'higher level' free energies?

I have seen 'free energy' arising from several contexts in very different forms, and each contains different amount of information.

For example

1. 
   

   free energy is defined as the logarithm of the partition function (which is a function of $T$), and the non-analyticity of the free energy at one point means phase transition happening at the corresponding temperature. Here for a given temperature free energy is a number.

   
   
   


2. 
   

   free energy is defined as Helmholtz or Gibbs free energy, here for a given temperature free energy is a function of some macroscopic parameter such as volume, pressure, etc. According to equilibrium phase transition theory, phase transition happens when two potential wells change stability (at some critical temperature $T_c$).

   
   


3. 
   

   In the field of protein folding simulation, there is also the so called 'free energy landscape', which is a 2D surface on some reaction coordinates (at a given temperature). Here the value of the function at certain point is given by the logarithm of the number of corresponding states counted during sampling.

   
   

My intuition for free energy is that it measures the stability of a configuration according to the combined effect of energy and entropy (number of possible states compatible with this configuration). But depending on how detailed information we want for a system, we can map it to a number (where all the possible stats of a system is viewed as one degenerate configuration), a 1D function, a 2D surface, etc. Is this the right way to understand the seemingly very different free energies in the above cases? Also, can anyone help to draw the connection between the phase transition in (1) and (2)? In particular, why the non-analyticity of the free energy implies phase transition?