* Classically, a generative model is one that describes a statistical process that generates a whole observation (maybe plus some hidden part) and thus gives you scores that are consistent between different observations. An example for this would be (unidirectional) language models, where the total probabilities for different possible sentences can be compared
* Classically, discriminative models are models where you also get a probability over outputs, but only conditional on some observed facts. As an example, a CNN transforms an image to logits that express the likelihood of different classes for this image. Hence, CNNs (used in this fashion) are a form of discriminative modeling.
If we map this to their "probabilistic families", Generative Models (in the old sense) would be Descriptive in their book, while Discriminative stays Discriminative.
The closest counterpart to their "Generative" models would be generative (in the old sense) models that assume a common distribution over hidden variables and a process that generates the actual output. However, this only loosely fits how GANs are used today, where you don't really assume a distribution over the hidden seed for an output, and where you get a single output (i.e. a deterministic function from seed to output instead of a distribution).
Neural networks have deterministic functions where MRFs and Gibbs sampling and all the pre-2012 machine learning goodies have probability distributions. It's not really helpful to use the term "probabilistic families" here when the most important bits of what they describe are non-probabilistic.