The following definition of vector fields were given in class,
A $\textbf{Smooth Vector Field on a manifold $M$}$ is a smooth function, $$s: M \to TM$$ such that $$\pi \circ s=id_M$$
We then went on to say that there are two alternative characterizations of vectors at a point. Below I am only presenting one of these alternative characterizations. That is;
We then defined $\textbf{a vector field on M}$ is an operator $D$ taking smooth functions and producing another smooth function that at every point measure the rate of change.
Are these definition equivalent? Furthermore, I am not sure what exactly these smooth functions are.
Yes, this is part of the standard game at the beginning of a manifolds course. The correspondence you should work out is that \begin{align*} s&\leftrightarrow D_s \\ s(x) &\leftrightarrow f\mapsto D_{s(x)}f \end{align*} where $D_{v_p} f$ is the directional derivative of $f$ in the direction of $v$ at $p$ (where each $v$ is a tangent vector at $p$).