I took a class this spring based on Phillip Moffitt's book: "Dancing With Life: Buddhist Insights for Finding Meaning and Joy in the Face of Suffering".
The book examines the 4 Noble Truths within the context of the 12 insights of the Buddha.. basically each of the Noble Truths is examined in the 3 insights, which sound very similar to the 3 types of understanding outlined above.
The first insight is an intellectual examination / understanding of the (external) teaching; the second is an ongoing examination from personal experience of what was determined in the first insight (and so a further intellectual understanding that comes internally from personal examination of one's experiences); and the third is a "knowing that you know" (the most difficult insight to attain).
Moffitt says the following about the third insight:
"You have also seen that your untrained mind with its endless stream of reactionary thoughts is unreliable. There is a reliable ground on which to stand to meet life that is your own awareness knowing itself. It is this ground of "knowing you know" that creates the steady, nonreactive state of mind that allows you to just be with life as it is.
A deep acceptance of life "just as it is" allows you to be more fully present in your life moment by moment, no matter how difficult or how sweet it is, and it empowers you to act more from your deepest values. Regardless of the circumstances of your life at any given time, your experience is richer, more alive. What's more, being with the dukkha opens you to an ever-deepening exploration of the inner life, eventually leading to what the Buddha calls "the deathless", which we will explore in the Third Noble Truth."
from page 66
I thought this would fit in nicely with the material you've presented here.