Standard basis in representation theory.

Question

Say we have a group $G=\{ g_1,g_2,...,g_3 \}$. In the permutation representation for $G$ acting on a set $A$, we define a basis by $$\tag{1} g\mathbf{e}_a=\mathbf{e}_{ga},\qquad a\in A, g\in G $$ Generally, I see people refer to these vectors as the standard basis, but others refer to the standard basis to satisfy eq. $(1)$ and written as $$\tag{2} \mathbf{e}_a=[1\quad 0\quad 0\quad ...\ ]^T,\quad \mathbf{e}_b=[0\quad 1\quad 0\quad ...\ ]^T,\quad ... $$ However, eq. $(2)$ is not the only way to write the vectors such that $(1)$ is fulfilled. For example, consider the regular representation of the cyclic group of order 3. Then we can select the vectors $$\tag{3} \mathbf{e}_\epsilon=[1\quad0\quad0]^T,\quad\mathbf{e}_g=[0\quad1\quad0]^T,\quad\mathbf{e}_{g^2}=[0\quad0\quad1]^T $$ with the representation $$ X(\epsilon)=\begin{bmatrix}1&0&0\\ 0&1&0\\ 0&0&1\end{bmatrix},\quad X(g)=\begin{bmatrix}0&0&1\\ 1&0&0\\ 0&1&0\end{bmatrix},\quad X\left(g^2\right)=\begin{bmatrix}0&1&0\\ 0&0&1\\ 1&0&0\end{bmatrix}. $$ But, we could also choose $$ \mathbf{e}_\epsilon=[1\quad1\quad1]^T,\quad\mathbf{e}_g=[1\quad e^{\frac{2}{3}\pi i}\quad e^{\frac{4}{3}\pi i}]^T,\quad\mathbf{e}_{g^2}=[1\quad e^{\frac{4}{3}\pi i}\quad e^{\frac{2}{3}\pi i}]^T, $$ with the representation $$ X(\epsilon)=\begin{bmatrix}1&0&0\\ 0&1&0\\ 0&0&1\end{bmatrix},\quad X(g)=\begin{bmatrix}1& 0 &0\\ 0&e^{\frac{4}{3}\pi i}&0\\ 0&0&e^{\frac{2}{3}\pi i}\end{bmatrix},\quad X\left(g^2\right)=\begin{bmatrix}1& 0 &0\\ 0&e^{\frac{2}{3}\pi i}&0\\ 0&0&e^{\frac{4}{3}\pi i}\end{bmatrix}. $$ In both cases, eq. $(1)$ is satisfied. So are they two ways of writing the standard basis, or is the standard basis defined to be the vectors given in eq. $(3)$?