When defining a manifold the domain and codomain of the transition maps is usually denote like this:
$$\varphi_\eta \circ \varphi_\lambda^{-1}: \varphi_\lambda(U_\lambda \cap U_\eta) \to \varphi_\eta(U_\lambda \cap U_\eta) $$
But say $\varphi_\lambda(U_\lambda) = V_\lambda$ and $\varphi_\eta(U_\eta) = V_\eta$ wouldn't it be shorter and therefore nicer to write
$$\varphi_\eta \circ \varphi_\lambda^{-1}: V_\lambda \cap V_\eta \to V_\lambda \cap V_\eta $$
Is there a reason why it is always denoted using the maps? (I have really never seen it using just the sets in $\mathbb R^n$)
It would be shorter, but wrong. :)
Consider two charts on the manifold $M = \mathbb R$. The domain of the first, $\phi_\eta$, is the interval $(0, 2)$, and the map takes $(0, 2)$ to $(0, 2)$ via $x \mapsto x$.
The domain of the second is $(1, 3)$, and the second takes $(1, 3)$ to $(101, 103)$ via $x \mapsto x + 100$.
Notice that the codomains are disjoint. Does that mean that $\phi_\eta \circ \phi_\lambda^{-1}$ has an empty domain? Not at all. The domain of this composite is $(101, 102)$, and the codomain is $(1, 2)$ (and indeed, the map is $x \mapsto x - 100$).