Let $F:M \to N$ be a smooth map between smooth manifolds and let $w\in T_s^0(N)$ be a $(0,s)$-tensor field on $N$. Then let us define the pullback of $w$ by $F$ as the $(0,s)$-tensor field $F^*w\in T_s^0(M)$ on $M$ defined, for any $p\in M$, by $$F^*w|_p := dF^*_p(w|_{F(p)})$$ where $$(\forall v_1,\dots,v_s \in T_pM) \quad dF^*_p(w|_{F(p)})(v_1,\dots,v_s) :=w|_{F(p)}(dF_pv_1,\dots, dF_pv_s).$$
Thus $F^* : T_s^0(N) \to T_s^0(M)$. Is there a way to show that this is actually just a specific case of the following generation of the pullback of $(r,s)$-tensor fields by $F$: $$F^* :=(F^{-1})_* : T_s^r(N) \to T_s^r(M)?$$
In your last line, you need $F^{-1}$ to be well-defined (at least differentiable), which is not the case in general.
The way you defined the pull-back operations is the most common one, and your last line should be the definition of push-forward when $F$ is a diffeomorphism, for example.