What is the difference between an ionic and molecular activity model? (Example: Diopside-Hedenbergite)
The activity of a chemical species is a key concept in thermodynamics of solutions. While the thermodynamic potential is the “true thermodynamic variable”, i.e. the one that plays a primary role in statistical mechanics, chemists do tend to like activity (which is univocally related to chemical potential) because it is close to another quantity which has clear physical meaning: concentration.

Now, relating the activity (or chemical potential) of a species to its concentration in a mixture is impossible to do in a general way. Models have been developed to provide approximations, however, the best known being the ideal solution, where activity equals molarity (concentration). Your “molecular mixing model” appears to be a model of ideal solid solution. However, one cannot expect this approximation to hold in case of strongly non-ideal solutions, and ionic solutions are prominent in that category. The “ionic mixing model” you describe thus appears to be another such approximation, more suited to these ions in a solid solution. I have never encountered it before, but I found a series of lecture slides where it seems to be very neatly introduced (see around slide 31).

While I cannot fully address the question, I hope this helps!