This "geometric" definition is well-known to category-theorists. See for example this youtube video by the Catsters, which introduces natural transformations. It should be also well-known to algebraic topologists working with model categories. But I have to admit that there are few introductions to category theory which emphasize this definition of a natural transformation.
Remark that this fits into a more general framework: For every category $C$, there is an isomorphism $[I,C] \cong Arr(C)$, where $Arr(C)$ is the arrow category of $C$. In particular, $Arr([C,D]) \cong [I,[C,D]] \cong [C \times I,D]$.