I think it is best to use no notation for change of basis! Let me explain what I mean.
One defines the matrix of a linear map $T:V \to W$ with respect to (ordered) bases $\{v_{1}, \dots, v_{n}\}$ and $\{w_{1}, \dots, w_{m}\}$ of $V$ and $W$ to be the $m \times n$ matrix whose $i$th column comprises the coordinates of $T(v_{i})$ with respect to the (ordered) basis $\{w_{1}, \dots, w_{m}\}$.
When $V = W$, the "change of basis matrix" is then simply the matrix of the identity transformation $\operatorname{Id}_{V}:V \to V$ with respect to the basis $\{v_{1}, \dots, v_{n}\}$ in the domain and $\{w_{1}, \dots, w_{n}\}$ in the codomain. More prosaically, the $i$th column of this matrix are the coordinates of $v_{i}$ in the (ordered) basis $\{w_{1}, \dots, w_{n}\}$.
Note that I have been fussily writing "ordered" in parentheses. This is because these notions do not depend simply on the basis (which is a set). They depend on the ordering of this basis implicit in the choice of indices. Reordering a given basis leads to a nontrivial change of coordinates! (By a permutation matrix). So when one speaks of the "change of basis" matrix one should really speak of the "change of ordered basis matrix". Of course no one does this, but this sometimes causes confusion for students, particularly if one writes something like ${}_{B}M_{C}$ and speaks of bases $B$ and $C$. Are $\{(1, 0, 0), (0, 1, 0), (0, 0, 1\}$ and $\{(1, 0, 0), (0, 0, 1), (0, 1, 0\}$ different bases? Not if one's definition of a basis is a linearly independent set that spans. But they are different ordered bases, and the usual conventions for change of basis matrices assign to them the permutation matrix
\begin{equation*}
\begin{pmatrix} 1 & 0 & 0\\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix}.
\end{equation*}
Notation like ${}_{B}M_{C}$ tends to obscure this subtle point.
More problematically, notation like ${}_{B}M_{C}$ is really quite sophisticated. Without going into details, this is essentially a transition function for the frame bundle (this explains why it satisfies cocycle identities like ${}_{B}M_{C} = ({}_{C}M_{B})^{-1}$ and ${}_{B}M_{C}{}_{C}M_{D} = {}_{B}M_{D}$).
Also, most of us can't remember whether ${}_{B}M_{C}$ means what it means or $({}_{B}M_{C})^{-1}$ (that is, on which side do $B$ and $C$ go? Equivalently, when do I write $A = P^{-1}BP$ and when $A = PBP^{-1}$?).
My solution is to use commutative diagrams. This sounds horrible, but my experience is that it works with first year engineering students. I draw diagrams whose vertices are vector spaces and edges are arrows between them labeled to indicate linear transformations. Choices of bases are indicated by writing them at each vector space and writing the label of the matrix determined by the linear transformation and the choices of bases below the corresponding edge. (I would show here what I mean but I can't figure out how to get xypic to work here.)
The relation expressing the change of the matrix of a linear transformation takes a form such as $PA = BP$ rather than $A = P^{-1}BP$ (or $B = PAP^{-1}$), where $P$ is the matrix of the identity transformation. There is no need to memorize a convention for placing inverses (equivalently the labels $B$ and $C$) because the correct choices are forced by the directions of the arrows.