But many other types of model would give you a joint distribution P(which digit, all pixels), so would be generative. Even if you only used it for classification.
https://en.wikipedia.org/wiki/Generative_model
I guess these days "generative" must mean "it is used to generate outputs that look like the training data".
But until recently, the meaning had to do with the information in the model, not how it's used.