As I learned, for a differential manifold with an affine connection $(M,\nabla)$, its torsion is defined as $$\begin{aligned} T:\mathscr{X}(M)\times \mathscr{X}(M) &\rightarrow \mathscr{X}(M)\\ (X,Y) &\mapsto T(X,Y):= \nabla_XY-\nabla_Y X-[X,Y] \end{aligned} $$ And I can check the multi-linearity.
My question is: Is $T$ a $(1,2)$ tensor (or tensor field)? I have learned that a (r,s) tensor field on $M$ is a smooth section of $T^{(r,s)}(M)$, mapping every $p\in M$ to a $(r,s)$ tensor over $T_p(M)$. But $T$ is defined globally, i.e. we cannot talk about $\nabla_{X_p}Y_p-\nabla_{Y_p} X_p-[X_p,Y_p]$.
So what does it exactly mean by saying $T$ a $(1,2)$ tensor? Are they equivalent definitions? Any comments or references are welcomed. THANKS!