Same evolution, different errors
We can always pick up an orthonormal basis |u_i\rangle in the environment and express the system–environment evolution as
\begin{aligned}
|\psi\rangle|e\rangle
\longmapsto &\sum_{ij} E_i|\psi\rangle|u_j\rangle\langle u_j|e_i\rangle
\\&= \sum_{j}\Big( \sum_i \langle u_j|e_i\rangle E_i\Big)|\psi\rangle|u_j\rangle
\\&= \sum_j M_j|\psi\rangle|u_j\rangle.
\end{aligned}
The new “error” operators M_j satisfy \sum_j M_j^\dagger M_j =\mathbf{1} and, in general, they are not unitary.
Now, the evolution of the density operator |\psi\rangle\langle\psi| can be written as
|\psi\rangle\langle\psi|\longmapsto \sum_j M_j|\psi\rangle\langle\psi| M_j^\dagger.
Which particular errors you choose depends of your choice of the basis in the environment.
If, instead of |u_j\rangle, you pick up a different basis, say |v_k\rangle, then
\begin{aligned}
|\psi\rangle|e\rangle
\longmapsto &\sum_j M_j|\psi\rangle|u_j\rangle
\\&= \sum_j M_j |\psi\rangle\sum_k|v_k\rangle\langle v_k|u_j\rangle
\\&= \sum_k \Big(\sum_j \langle v_k|u_j\rangle M_j \Big)|\psi\rangle|v_k\rangle
\\&= \sum_k N_k|\psi\rangle|v_k\rangle,
\end{aligned}
and, consequently,
|\psi\rangle\langle\psi|\longmapsto \sum_k N_k|\psi\rangle\langle\psi| N_k^\dagger.
The new “errors” satisfy \sum_k N_k^\dagger N_k = \mathbf{1}, and the error operators N_k and M_j are related by the unitary matrix U_{kj}=\langle v_k|u_j\rangle.