Does this definition of a pullback of a differential form make sense?
$\phi:M\rightarrow N$ and $\alpha\in\Omega^r(N)$ then define $$(\phi^*\alpha(X_1,\dots,X_r))(p) := \alpha(\phi(p))(\phi_{*_p}X_1(p),\dots,\phi_{*_p}X_r(p))$$ If it does make sense, it is annoying because it mixes both interpretations of a differential form. Ie on one side it is a map from vector fields to smooth functions and on the other, it is a map from a manifold to tangent bundle.
If you want to view both sides consistently, use
$$(\phi^*\alpha)(p)(X_1(p), \dots, X_r(p)) = \alpha(\phi(p))(\phi_{*_p}X_1(p), \dots, \phi_{*_p}X_r(p)).$$