“Although I didn’t think the distinction was so important to focus on in this essay.” The distinction is important.
Do you agree that in logic, A |- A says something akin to “from a true A (a non-logical axiom), you may infer the truth of A”? If so, I don’t see any circularity there; anything on the left hand side of the |- is an ‘additional’ axiom of your theory, hence accepted as true. Circular reasoning, by my interpretation, is where you infer/assume as true what you’re in the process of trying to prove to be true. This is not the case with A |- A, in which the occurrence of A on the left of the |- means you are taking it as true to begin with. That’s why I assumed you meant A → A when you said “A implies A”.
The law (A → A) → A would allow you to infer A from A → A, which would, indeed, allow you to infer A without it having being established as true, doing so, rather, solely from the premise that if it were true, it would be true, thereby concluding that it is true. This is closer to the interpretation of circular reasoning I gave above.
“It also doesn’t fit the geometric intuition behind ‘circular’ well. leads back around to where it started”. I think this appeal to geometric intuition is misleading, which might explain the confusion. All A |- A it is saying is that if you already know/believe A, then you may infer it. In a formal system it may take several steps to derive (‘lead to’) A from A, but I don’t think that circular reasoning should be tied to proof theory.
I think the issue might be that I’m interpreting circular reasoning as something stronger than you; ie, in the pernicious sense which explains why “The idea that circular reasoning is bad is widespread”.
I suspect that according to your interpretation all valid deductive reasoning is circular in some way, circularity thus being necessary for valid deductive reasoning. In this regard, circularity would be a desirable attribute.
In contrast, my interpretation is one in which in the process of affirming a belief, one presupposes something that would require to have already affirmed the same belief; what is sometimes called “begging the question”.
In this context, I don’t regard A |- A (circular in your sense, but not in mine) as problematic, as it just involves inferring something that has already been affirmed.