Here is a page from Guillemin-Pollack's differential topology:
My question is: At the bottom he defines $df=d\psi\circ dh\circ d\phi^{-1}$. Why doesn't he just define $df=dh$, like here: http://www.math.toronto.edu/mat1300/smoothmaps.4.pdf? It makes computations easier. Does one imply the other?
You can't say that $df=dh$ since the domains are different. Notice that $df_{(x,y)}$ is a linear transformation from $T_{p}(x)$ to $T_{y}(Y)$ while $dh_{(0,0)}$ is a linear transformation from $\mathbb{R}^{k}$ to $\mathbb{R}^{t}$.
You define $df$ using the coordinate charts that you would need if you want to do explicit calculations. However, you can prove that the definition doesn't depend on what coordinate chart you choose.
I can't see the definition you suggest in your link, but see that at the beginning $G$ is defined as a composition, $G= \psi\circ F\circ\phi^{−1}$, which gives you the coordinate expression for $F$. Then you can do everything using $G$ and $dG$ since the results are independent of the coordinates.