Consider a complex surface $S$. An isolated singularity of $S$ is called a rational double point if it is locally analytically isomorphic to $(f^{-1}(0)\subset \Bbb C^3,0)$ where $f=f(x,y,z)$ is one of the following polynomials: $x^{k+1}+y^2+z^2 ~(k\geq 1), x^{k-1}+xy^2+z^2~(k\geq 4), x^3+y^4+z^2, x^3+xy^3+z^2, x^3+y^5+z^2$.
An isolated singularity of $S$ is called a quotient singularity if it is locally analytically isomorphic to $(\Bbb C^2/G,0)$ where $G$ is a finite subgroup of $\text{GL}(2,\Bbb C)$.
According to Theorem A of the paper "Fifteen characterizations of rational double points and simple critical points" by Durfee (link: https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.474.1363&rep=rep1&type=pdf), a rational double point is a quotient singularity and conversely (Characterizations A1 and A5 in the paper).
Also, in chapter A.3 in the paper, it says that rational double points has the following property: the exceptional curves in the minimal resolution are rational curves with square -2.
On the other hand, for example in this link: https://www.homepages.ucl.ac.uk/~ucahjde/blog/cyclic1.html, there is a different result of the minimal resolution of a (cyclic) quotient singularity.
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Best Answer
The issue is with asking $G$ to be a subgroup of $GL(2,\Bbb C)$. The correct definition in order to get the Du Val singularities (what you're talking about in the first portion of the problem) is that $G$ should be a subgroup of $SL(2,\Bbb C)$ - you'll note that in general, the matrices defining the group action in the second link are not in $SL(2,\Bbb C)$ and only in $GL(2,\Bbb C)$.