So you have a non minimal DFA $M$ and you create a minimal DFA $M_{/≈}$. What is the meaning of $M_{/≈}$? I understand that the $≈$ tells that this $M_{/≈}$ is the equivalent version of the original $M$ but I see no specific reason to use $/$ instead of something akin to $M`_≈$
And if there is a reason, what is the name of the $M_{/≈}$ notation so I can look further into it.
If $\sim$ is an equivalence relation on a set $S$, then the “quotient set” $S/{\sim}$ is the set of equivalence classes (Wikipedia).
In your case, it's $M/{\approx}$ (not $M_{/{\approx}}$), the set of equivalence classes under the relation $\approx$ on $M$. It's unclear what you mean by “the equivalent version of the original $M$”, but that seems to be a misinterpretation.