There is at least one more important example where people who believe that "differential forms are used only in problems where they already appear in basic definitions" are completely wrong. Differential forms and the exterior differential were created (or discovered) by Elie Cartan in the 1890's in order to solve the Pfaff problem, as follows. Let us think in $R^n$. Certainly these people will admit that a differential form of degree one, that is, a covector field $$\omega=f_1dx_1+\dotsb+f_ndx_n$$ is a natural object and deserves some attention…. Here $f_1, \dotsc, f_n$ are real functions of $x_1, \dotsc, x_n$. A very natural and universal method in sciences, in order to study an object, is to consider it in the coordinates system where it gets the simplest form…. Pfaff's problem is: given such a form $\omega$, in a small neighborhood of a given point, can I reduce the number of variables to some $k<n$? In other words, can I find local curvilinear coordinates $y_1, \dotsc, y_n$ such that $$\omega=g_1dy_1+\dotsb+g_kdy_k$$ where $g_1, \dotsc, g_k$ are real functions of $y_1, \dotsc, y_k$? In order to solve this, Cartan defines the exterior differential $2$-form $d\omega$, and considers what he calls the successive differentials of $\omega$, namely: \begin{align*} & \omega'=d\omega, \\ & \omega''=\omega\wedge d\omega, \\ & \omega'''=d\omega\wedge d\omega, \dotsc. \end{align*}
Generally, he defines $\omega^{(k)}$ as $(d\omega)^{(k+1)/2}$ (using the wedge product) if $k$ is odd, and as $\omega\wedge(d\omega)^{k/2}$ if $k$ is even. He proves that the exterior differential $d$ is invariant under coordinates changes (quite a discovery!) and concludes that a necessary condition for the $k$-reducibility is that $\omega^{(\ell)}$ vanishes for every $\ell>k$. Then, his real work is to prove that this necessary condition is sufficient, in the real-analytic case (the smooth case requires some more subtle nondegeneracy condition, if I'm correct).
Also note how these simple formulas contain a lot of geometry. Foliations, symplectic and contact geometries, appear here in the consideration of the (non)vanishing of the successive derivates. Differential forms were indeed one of Elie Cartan's great discoveries.
