Visually, the rotation part makes it much less intuitive than writing the result matrix next to the first operand and the second operand above the result matrix
I agree. Matrix multiplication can pretty succinctly be defined in terms of the dot product between the row and column of the inputs at the corresponding spot in the output. When I first learned I already new a little about the properties of dot products so it made the “magical” properties of matrix multiplication more understandable. Maybe not the way a mathematician would define it — they might look at dot products as a special case of matrix multiplication thus teach the latter first — but it was at least intuitive.
I think mathematicians will usually be aware of many ways of defining something. Because different definitions will suggest different approaches to a problem, and suggest different ways to generalize a concept.
Yours is an important one.
It can be nice to help visualize, though, for example if you have a block matrix and you know some blocks have particular properties, it can be helpful to imagine the steps of an algorithm as various blocks sliding across each other.