I think they are here to show a particular thing of interest, because they're all very similar. There is always a category for contexts whose pullbacks correspond to substitutions, there's always a functor that tells you the types under a certain context, etc., the only difference is that they are described in different perspectives. For example, in CwA you have display maps anyway, and if you start the definition with display maps you get a display map category. If you make some sets isomorphic to global sections of display maps you get a category with families.
In some context, as models of type theories, these constructions are all equivalent modulo technical details, because they are just different ways to describe the same thing. If your model lacks a property needed by the type theory, you'll have to add it, if your model has extra properties not needed by the type theory, it is better to ignore them because they may have consequences (for example, if you prove a theorem about your type theory using a cat model, it may not be true in your type theory but you proved it using the extra properties. You don't want that to happen). So essentially they're equivalent in this sense.
The difference arise when you actually want to do something with the categorical structures themselves, and you can imagine that some definitions are convenient in some sense but not others, etc., so we tend to use the most convenient ones.