The description you posted from the guide already has the answer, but you seem to be glossing over that. The priorities apply in Self Service, not anywhere else. The images above are from the JSS, so that's why they aren't ordering by their priority number there. But to note, you could click on the Priority column header to sort your categories in their numerically assigned priority order.

It might be nice if we could have the Category section in the JSS default to sorting by priority number, but it currently doesn't allow for that.

I guess I just had it in my mind that no matter where it is displayed it would be in that priority order. Thanks for pointing it out for me, I apparently kept missing the self-service only part. My thought process wants everything to show the exact same way whether its user facing or admin facing.

Not an unreasonable expectation. If I were you I would file a Feature Request asking for the JSS to respect priority order, for Categories or any other items that use them, rather than relegating the feature to only Self Service. Such a feature request will likely fall into a more broad bucket of requests to make organizing of data in the JSS more flexible and intuitive. Its a little all over the place right now to be honest.