Let $R$ be a finite commutative principal ideal ring. Let $n$ be a positive integer. For $i=1, \ldots, n-1$ we let \begin{align*} w_i := w_i(x_{i+1}, \ldots, x_n) = \sum_{j=i+1}^n t_{ij}x_j \in R[x_{i+1}, \ldots, x_r], \end{align*} where $t_{ij}$ are some fixed elements of $R$, for $1 \leq i < j \leq n$. Let $m_1, \ldots, m_{n-1}$ be arbitrary elements of $R$ and let $a_1, \ldots, a_{n-1}$ be such that $a_im_i = 0$, that is, $a_i$ is the annihilator of $m_i$. For $i=1, \ldots, n$ we set \begin{align*} y_i = m_ix_i + w_i(x_{i+1}, \ldots, x_n), \end{align*} where for convenience we let $y_n = x_n$.
Consider the $R$-module of $R^n = \{(x_1, \ldots, x_n) | x_i \in R\}$ where addition is component wise and the action of $R$ on $R^n$ is scalar multiplication. Define $\phi_i: R^n \rightarrow R^n$ by \begin{align*} \phi_i(x_1, \ldots,x_i, \ldots, x_n) = (x_1, \ldots,y_i, \ldots, x_n). \end{align*}
Then for each $i$, we set $K_i:= \ker(\phi_i) = \{0\} \times \cdots \times <a_i> \times \cdots \times \{0\}$. By the first isomorphism theorem, we have \begin{align*} \phi_i(R^n) \simeq R^n/K_i \simeq R \times \cdots \times R/<a_i> \times \cdots \times R \end{align*}
I claim that $R/<a_i> \simeq <m_i>$. Supposing the claim holds, this implies \begin{align*} \phi_i(R^n) \simeq R \times \cdots \times <m_i> \times \cdots \times R. \end{align*}
But then if I define $\psi_i: R^n \rightarrow R^n$ by \begin{align*} \psi_i(x_1, \ldots, x_i, \ldots, x_n) = (x_1, \ldots, m_ix_i, \ldots, x_n) \end{align*} then by similar reasoning it will follow that \begin{align*} \psi_i(R^n) \simeq R \times \cdots \times <m_i> \times \cdots \times R \end{align*}
Both $\phi_i$ and $\psi_i$ are elements of $Hom_R(R^n,R^n)$, that is, the ring of $R$-module endomorphisms on $R^n$. In fact, they will be equal when defined on the $i^{th}$ coordinate submodule $\{(0, \ldots, x_i, \ldots, 0) | x_i \in R\}$. But $\phi_i$ and $\psi_i$ aren't equal maps.
So when am I justified in using $\psi_i$ instead of $\phi_i$? The images of $R^n$ under both maps are isomorphic as $R$-modules. Is there much difference in using one map over the other? Will these maps belong to some sort of equivalence class? Or a coset of $Hom_R(R^n,R^n)$ or something?
I'm looking at a question where I want to replace $\phi_i$ by $\psi_i$, but I don't know if I'm losing information by doing so. See previous question here. Any help on how $\phi_i$ and $\psi_i$ are related would be great!