Given the tangent bundle $\pi:TM \rightarrow M$, a generic vector field $X \in T_{0}^{1}(TM)$ can be written in local coordinates as: \begin{equation} X = A^a(x,v)\frac{\partial}{\partial x^a} + B^a(x,v)\frac{\partial}{\partial v^a}. \end{equation} The (smooth) projection function $\pi$ induces a map $T\pi:TTM\rightarrow TM$, defined pointwise as $(T\pi)(X):=\pi_*X$, where $X$ is to be interpreted as a point in $TTM$ on the left hand side, and as the corresponding vector at the point $(x,v)\in TM$ on the right hand side.
Now, my problem is: if $X$ is a vector field on $TM$, does $\pi_*X$ define a vector field on $M$?
It seems to me that this is only possible if \begin{equation} X = A^a(x)\frac{\partial}{\partial x^a} + B^a(x,v)\frac{\partial}{\partial v^a}. \end{equation} that is if $A^a$ does not depend on $v$. In that case, $\pi_*X=A^a(x)\frac{\partial}{\partial x^a}$, which is a well-defined vector field on M.
Nonetheless, in all the manuals I have consulted this point is not discussed at all and it is simply stated that $\pi\circ\pi_{TM} = \pi\circ T\pi$, where $\pi_{TM}$ is the projection map of the second tangent bundle $\pi_{TM}:TTM\rightarrow TM$.
Any insight?