Three types of double proton transfer

I previously posted about how double proton transfer is a concrete example of a chemical reaction that can occur via either a concerted or sequential process?
Precisely defining this question and answering it is a subtle issue.

There is a nice classification of types of potential energy surfaces for double proton transfer, summarised on the website of Antonio Fernandez-Ramos. It is based on a very simple model potential energy surface described here and compared to surfaces from computational quantum chemistry [at the DFT level] here [source of the figures below].

There are three qualitatively different potential energy surfaces, depending on the strength of the coupling of the motion of the two protons.

(1) One transition state and two minima, as in the formic acid dimer;

(2) Two equivalent transition states, one maxima and two minima, as in the pyrazole dimer;

(3) Four transition states, one maxima and four minima, as in porphine.

I am pretty happy because I am developing a simple diabatic state model that captures all of the above cases.