This is a follow up to the answer to this question: Using Burnside's lemma for a triangle
I am confused about this because $D3$ is the set of symmetries of the triangle $D_3=S_3=\{g:V=\{1,2,3 \} \rightarrow V \mid g \text{ is bijective}\}$ i.e. permutations of $V$. $X$ is the set of colorings, $X=\{x: V \rightarrow \{R,B,Y,G\}\}$, say for four colors.
The notation the answerer used, g.x, implies a permutation in $D3$ acts on the coloring of the vertices. However, $g(x(v))$ for $x$ in $X$, is not a valid composition since g’s domain isn’t $\{R, G, Y, B\}$.Wouldn’t it be more accurate to talk about the action of $x$ on $g$?
You are confused about the action. The group $G$ is not acting on the set of colors, it is acting on the set of colored triangles. In the problem you linked to, it is the edges that are colored, using four possible colors (not the vertices).
Imagine a triangle on the plane, with vertices at $(1,0)$, $(-\frac{\sqrt{3}}{2},\frac{1}{2})$, and $(-\frac{\sqrt{3}}{2},-\frac{1}{2})$, in that order.
Denote a coloring by listing three colors in a given order, with the first color corresponding to the edge from $(1,0)$ to the vertex in the second quadrant, the second color corresponding to the edge going from the second to the third quadrant, and the third color corresponding to the edge going from the third quadrant to $(1,0)$. Label the colors $1$, $2$, $3$, $4$. So a coloring is a three digit number all of whose digits are between $1$ and $4$.
Denote $D_3$, as usual, by $D_3=\{e, r, r^2, s, rs, r^2s\}$, where $e$ is the identity rigid motion, $r$ is a counterclockwise rotation by $120$ degrees, $r^2$ is a counterclockwise rotation by $240$ degrees, $s$ is the reflection about the $x$-axis, $rs$ is the reflection followed by the rotation $r$, and $r^2s$ is the reflection followied by the rotation $r^2$.
The group $D_3$ does not act on the colors, it acts on the full colorings. The set of colorings consists of $4^3 = 64$ colored triangles: $$111, 112, 113, 114, 121, 122, 123, 124, 131,\ldots, 441, 442, 443, 444.$$
Let $C$ be the set of colorings. Then $D_3$ acts on $C$ by taking a colored triangle, applying to it an element of $D_3$, and obtaining a colored triangle. For example, if you take the colored triangle $123$, then $$\begin{align*} e\cdot 123 &= 123\\ r\cdot 123 &= 312\\ r^2\cdot 123 &= 231\\ s\cdot 123 &= 321\\ rs\cdot 123 &= 132\\ r^2s\cdot 123&=213 \end{align*}$$ We can then see that the orbit of the coloring $123$ is precisely the colorings $\{123, 132, 213, 231, 312, 321\}$.
You can instead think of the colorings as maps $c\colon\{e_1,e_2,e_3\}\to \{1,2,3,4\}$, where $c(e_i)$ is the color of the $i$th edge. To describe a coloring, then, I need to tell you its value at each of $e_1$, $e_2$, $e_3$.
So if $c$ is a coloring, and $g\in D_3$, then to describe $g\cdot c$ I can tell you the value of $g\cdot c$ at each of $e_1$, $e_2$, $e_3$. So what you want is to describe wht $(g\cdot c)(e_1)$, $(g\cdot c)(e_2)$, and $(g\cdot c)(e_3)$ are.
You are mistakenly writing these expressions as $g(c(e_1))$, $g(c(e_2))$, and $g(c(e_2))$. You are correct that those expressions are nonsensical... but those expressions are not in the original post, they are in your incorrect paraphrase of what is written in that post.