Let $M$ be a manifold, then a 1-form $\omega$ is a function that assigns to each point $p \in M$ a linear function or covector. This definition seems natural to me as it exactly captures a 1-form being a section of the cotangent bundle.
On the other hand, we may define the differential of a function $f \in C^\infty(M)$ to be the 1-form: $$(df)(X) = Xf$$ where $X$ is a vector field.
What is confusing to me is that as the 1-form are now defined over the space of vector fields (or equivalently, the set of all smooth sections of $TM$) and not $M$.
If we specify both a tangent vector and a point then it seems these two definitions are equivalent. The first definition first uses $p \in M$ to define the linear function $\omega_p$, and then evaluates it at some $v \in T_pM$ specified by the vector field $X$. The second definition works the other way around, where $X$ first fixes the tangent vector $v$ and then we evaluate at $p$.
Are these definitions both correct and hence equivalent, or is one an informal way or saying the other?