"Represent" vs "stand for" vs "means"

I am writing an academic paper.
I wrote the following sentences

Notation A ↔ B stands for that A corresponds to B. Qualid and ident
stand for qualified identifiers and simple identifiers,
respectively.

In the above example, I used "stand for" twice. My friend suggested me to replace the first
"stand for" with "represent".
Thus, I should use "Notation A ↔ B represents that A corresponds to B".
However, he told me that using "stand for" for "Qualid and ident" made no problem.

I cannot understand the reason why "Notation A ↔ B stands for that A" is unsuitable.
Moreover, I am confused with the reason why "Qualid and ident stand for" is fine.

Thanks in advance.