To clarify, I interpret discourse as the process of expressing an interpretation of an understanding rooted in the intelligibility of being-in-the-world. As you can see, the definition does not require that an expression be either public or communicative in any form, including verbal/linguistic forms (though it can be.).
And for those who are uncomfortable with the notion of a private discourse divorced from the verbal and/or linguistic, I am happy to interpret "[X} as the process of expressing an interpretation of an understanding rooted in the intelligibility of being-in-the-world".
If we had no understanding of the world, we would have no basis upon with to choose one act over another or several others. But we do have an understanding of the world and in order to act with purpose there are times we must render "explicit" our understanding. For example and if one wants a
good potato, it is not enough to understand that heat is useful for baking potatoes. One must understand how much heat and for how long, i.e., one's understanding must be more
explicit than heat is useful.
And the process of rendering explicit our understanding is what Heidegger calls
interpretation. In turn, discourse is an expression of our interpretation. And I express my interpretation of my understanding of what is necessary to bake a good potato by turning the oven to 425 degrees and setting the timer to 30 minutes at which point I would turn the potato and reset the timer for another 30 minutes.
And none of that would be possible absent a general understanding of cooking and a more explicit understanding of cooking a potato and the ability to put that understanding in to actions both verbal and non-verbal and both private and public.
Keep in mind, Heidegger is attempting to describe our regular and ongoing involvement in the world in our
average every dayness, i.e,, walking the dogs, mowing the lawn, taking out the trash, baking potatoes. And very little of our average everyday involvement in the world is intended to be publicly communicative.