Don’t know the meaning of “more accurate” here as these are just the perspectives as you said.
I don’t know how far you have gone through with the first perspective because precisely you can say a “function” is transforming a vector into another another vector. Because here, function means “Linear Transformation” and every matrix is a representation of a linear transformation.
Suppose, you are working in the real domain and when you write:
$\begin{pmatrix} a &b \\ c &d \end{pmatrix}\begin{pmatrix} x\\y \end{pmatrix}=\begin{pmatrix} e\\f \end{pmatrix}$
It means you are transforming a two-dimensional vector into another two-dimensional vector according to the rule:
$T: \mathbb{R^2} \rightarrow \mathbb{R^2}$ with
$T(x,y) = (ax+by, cx+dy) = (e,f)$
