I will reply to this in the sense of
"do you believe you are aware of the inferential connections between your expertise and layperson-level knowledge?",
since I am not so familiar with the formalism of a "Level 2" understanding.
My uninteresting, simple answer is: yes.
My philosophical answer is that I find the entire question to be very interesting and strange. That is, the relationship between teaching and understanding is quite strange IMO. There are many people who are poor teachers but who excel in their discipline. It seems to be a contradiction because high-level teaching skill seems to be a sufficient, and possibly necessary condition for masterful understanding.
Personally I resolve this contradiction in the following way. I feel like my own limitations make it to where I am forced to learn a subject by progressing at it in very simplistic strokes. By the time I have reached a mastery, I feel very capable of teaching it to others, since I have been forced to understand it myself in the most simplistic way possible.
Other people, who are possibly quite brilliant, are able to master some subjects without having to transmute the information into a simpler level. Consequentially, they are unable to make the sort of connections that you describe as being necessary for teaching.
Personally I feel that the latter category of people must be missing something, but I am unable to make a convincing argument for this point.
A lot of the questions you pose, including the definition of the Level 2 formalism, are addressed in the article I linked (and wrote).
I classify those who can do something well but not explain or understand the connections from the inputs and outputs to the rest of the world, to be at a Level 1 understanding. It's certainly an accomplishment, but I agree with you that it's missing something: the ability to recognize where it fits in with the rest of reality (Level 2) and the command of a reliable truth-detecting procedure that can "repair" gaps ...
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