Let's look at the etymology.
Old English understandan "to comprehend, grasp the
idea of, receive from a word or words or from a sign the idea it is
intended to convey; to view in a certain way," probably literally
"stand in the midst of,
"the under is not the usual word meaning "beneath," but from Old
English under, from PIE *nter- "between, among" (source also of
Sanskrit antar "among, between," Latin inter "between, among," Greek
entera "intestines;" see inter-)
Perhaps the ultimate sense is "be close to;" compare Greek epistamai
"I know how, I know," literally "I stand upon."
The intransitive sense of "have the use of the intellectual faculties;
be an intelligent and conscious being" also is in late Old English. In
Middle English also "reflect, muse, be thoughtful; imagine; be
suspicious of; pay attention, take note; strive for; plan, intend;
conceive (a child)." Also sometimes literal, "to occupy space at a
lower level" (late 14c.) and, figuratively, "to submit."
-excerpts from the Etymonline entry
"expound the meaning of, render clear or explicit," from Old French
interpreter "explain; translate"rom Latin interpretari "explain,
from interpres "agent, translator," from inter "between" (see inter-)
& second element probably from PIE *per- (5) "to traffic in, sell."
I find the distinction interesting. How do you demonstrate understanding? Often, with an interpretation. Did you understand eg second order partial dfferential equations? Then demonstrate that by interpreting this specific example to extract useful information from this case of the form.. &c &c
Or like in Zen, with the adding of a capping verse to a koan - which illustrates how understanding can be something transcendental, non-verbal, yet given a paeticular form and expression, in a specfic interpretation of a problem: a momentary relative expression of the most general, once that has been grasped.
I would suggest that understanding what understanding is, is like finding the meaning of meaning. It involves a recursion of the process itself, to make sense of it. And, manifesting an interpretation, is one concrete method of demonstrating an understanding.
Im case 201 of Dogen's Shobogenzo:
"Venerable Bodhidharma was about to go back to India. He said to his
students, "The time has come. Can you express your understanding?" One
of the students, Daofu said, "My present view is that we should
neither be attached to letters, nor be apart from letters, to allow
the Way to function freely." Bodhidharma said, "You have attained my
skin." Nun Zongchi said, "My view is that it is like the joy of seeing
Akshobhya Buddha’s land just once and not again." Bodhidharma
said, "You have attained my flesh." Daoyu said, "The four great
elements are originally empty and the five skandhas do not exist.
Therefore, I see nothing to be attained." Bodhidharma said, "You have
attained my bones." Finally Huike came forward, made a full bow, stood
up, and returned to where he was. Bodhidharma said, "You have attained
my marrow." Thus he transmitted the Dharma and robe to Huike.
So we have four interpretations, manifesting different understandings, of one realisation.
I would look to Wittgenstein's advice:
"In this sort of predicament, always ask yourself: How did we learn
the meaning of this word ("good", for instance)? From what sort of
examples? In what language-games? Then it will be easier for you to
see that the word must have a family of meanings."
-from Philosophical Investigations
How do we use meaning? How interpretation? These are language-games. These are embedded in modes of life. How do we teach them? What purposes do we put them to? In that examination, is your answer. And, it isn't fixed. The domain of understanding, shifts, expands, becomes more convoluted and interconnected, sometimes simplifies. It is not 'dead facts'.
Other modes to demonstrate understanding:
1 to show, manifest, or prove, esp. by reasoning, evidence, etc.
"it is easy to demonstrate the truth of this proposition" 2 tr to evince; reveal the existence of the scheme later demonstrated a
fatal flaw 3 tr to explain or illustrate by experiment, example,
etc. 4 tr to display, operate, and explain the workings of (a
machine, product, etc.) 5 intr to manifest support, protest, etc.,
by public parades or rallies 6 intr to be employed as a
demonstrator of machinery, etc. 7 intr (Military) to make a
show of force, esp. in order to deceive one's enemy