In general, given a map of smooth manifolds $\phi:N\to M$, derivations $C^\infty M \to C^\infty N$ are in canonical one to one correspondence with sections of the pullback bundel $\phi^* TM$ on $N$. Such a section is a smooth map that associates to each point $p\in N$ a tangent vector in $T_{\phi(p)}M$. These are sometimes called vector fields relative to the map $\phi$ and can also be pictured as infinitesimal deformations of the map $\phi$. In your case $N$ is the tangent bundle of $M$ with its canonical projection.

In general, given a map of smooth manifolds $\phi:N\to M$, derivations $C^\infty M \to C^\infty N$ are in canonical one to one correspondence with sections of the pullback bundel $\phi^* TM$ on $N$. Such a section is a smooth map that associates to each point $p\in N$ a tangent vector in $T_{\phi(p)}M$. These are sometimes called vector fields relative to the map $\phi$ and can also be pictured as infinitesimal deformations of the map $\phi$. In your case $N$ is the cotangent bundle of $M$ with its canonical projection.