Could you please tell me if my understanding is correct?
Coordinates:
Consider the vector space $K^n$ and define the standard basis to be: $\underline{S}=\{e_1,\dots,e_n\}$, where $e_i$ has a single $1$ in the $i^{\text{th}}$ position.
Any vector written $(a_1,\dots,a_n)_{\underline{S}}=\sum a_ie_i$ is said to be in standard coordinates. If we define a second basis $\underline{V}=\{v_1,\dots,v_n\}$ of $K^n$, then a vector $(a_1,\dots,a_n)_{\underline{V}}=\sum a_iv_i$ is said to be in $\underline{V}$ coordinates.
Change of basis:
We can define a map $\varphi:K^n\to K^n$ by letting $\varphi(v_i)=e_i$ and extending linearly. This will take points in $\underline{V}$ coordinates to points in $\underline{S}$ coordinates. In particular we can write the change of basis matrix for this linear map: $$[\varphi]=[\underline{V}\to \underline{S}]=([v_1]_{\underline{S}}\cdots[v_n]_{\underline{S}}),$$ where $[v_i]_{\underline{S}}$ is the column vector of $v_i$ written in $\underline{S}$ coordinates.
Yes, it is correct. The notations are not standard, but the content looks good.