Here is the definition for a pullback induced by a function $f: N \rightarrow M$ on a generic vector bundle $\pi: E \rightarrow M$:
$$f^∗(E) = \{ (p,e) \in M×E \ | \ f(p) = \pi (e) \}$$
For instance, this means that the fibers of the pullback of the bundle of alternating forms $(f^* (\Lambda M))_p$ can be identified with $\Lambda _{f(p)} M = \text{Alt} (T^* _{f(p)} M)$.
The pullback of a differential form is then naturally defined as
$$f^* \omega : p \mapsto (p, \omega \circ f) = (p, f(p), \omega_{f(p)})$$
where $ \omega_{f(p)} \in \text{Alt}(T ^* _{f(p)} M ) $ so that $f^* \omega$ is a section of $f^∗ (\Lambda M)$.
But then I've came across a definition specific for differential forms:
$$f^*\omega : p \mapsto (p, \omega_{f(p)} \circ d_p f)$$
Where the composition is intended for each input vector. This clearly is no longer a section of $f^∗(\Lambda M)$ but a section of $\Lambda N$, as $ \omega_{f(p)} \circ d_p f \in \text{Alt}(T ^* _{p} N ) $, so these two are not consistent.