Intuitive understanding of the matrix of a linear transformation

Yes, that's a great way to think about it.

The matrix is like the minimal information that you need to write down in order to fully specify the linear transformation.

Here is a bit of a more fleshed-out version of what you're saying, from my perspective.

What does it mean to take a basis of a finite-dimensional vector space $V$? It means that you identify each vector $v \in V$ uniquely with a tuple $(v_{1},\ldots,v_{n})$ of components. Now, linear maps work directly with your vector space. Consider the linear map "rotate anticlockwise by $\pi/2$" in the plane. The map picks up an arrow (or however you prefer to think of a vector) and physically rotates it round.

On the other hand, we can identify the plane with $\mathbb{R}^{2}$ via the standard basis, and then the rotation matrix $$R=\left(\begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array}\right)$$ tells you how to change the components of your vector. If you picked a different basis, then your components would change differently. Precisely, if you change your basis with the matrix $P$, then your new transformation matrix would be $$R'=P^{-1}RP$$

In summary, linear maps change your vectors, while matrix multiplication changes your components.