Linear connection on a manifold: Math vs. Physics

I have learned some Riemannian Geometry in a strongly mathematical framework, precisely from the book "J.M.Lee - Riemannian Manifolds: An introduction to Curvature". Now I'm trying to learn Relativity from the Wald's book, but I have many problems to match the Riemannian Geometry notions from the mathematical framework to the physical one.

Consider the notion of linear connection: For me a linear connection $\nabla$ is a function $$\nabla:\mathcal T(M)\times\mathcal T(M)\longrightarrow\mathcal T(M)$$ $$(X,Y)\longmapsto \nabla_XY$$ where $\mathcal T(M)$ is the $C^\infty(M)$-module of sooth vector fields (sections of the tangent bundle). This function $\nabla$ has certain properties that allow to write $\nabla_XY$ in local coordinates. I know that exists an essentially unique way to define a (Koszul) connection $\overline\nabla$ on an tensor field starting from $\nabla$ and with the connection $\overline\nabla$ I can define a total covariant derivative for tensor fields. All these reasonings are done without computations in coordinates, but using strongly the "Tensor Characterization Lemma".

Wald instead says that a covariant derivative is a way to associate to a tensor field $T\in\mathcal T^{(k,\ell)}(M)$ another tensor field $\nabla T\in T^{(k,\ell+1)}(M)$ written in the index notation as $$\nabla_c{T^{a_1,\ldots,a_k}}_{b_1,\ldots,b_\ell}$$ such that it satisfies certain conditions. Now even if I understand the abstract index notation (infact I recognize that $\nabla_c{T^{a_1,\ldots,a_k}}_{b_1,\ldots,b_\ell}$ is a $(k,\ell+1)$-tensor) I don't understand how this approach coincides with the above one.

Your first definition of $\nabla$ often is referred to as covariant derivative of a vector field. More general, you can define a connection $\nabla$ on a vector bundle $E$ as a map between sections on $E$ $$\nabla:\Gamma(E)\to\Gamma(E\otimes T^*M)\qquad\text{satisfying}\qquad\nabla(fe_1+e_2)=e_1\otimes\mathrm{d}f+f\nabla e_1+\nabla e_2$$ for $e_i\in\Gamma(E)$ and $f\in C^\infty(M)$. This is Wald's approach.
On the other hand, if you "insert" a vector field $X\in\mathcal{T}(M)$ into the second component, you obtain a map $$\nabla:\Gamma(E)\times\mathcal{T}(M)\to\Gamma(E)$$ which is commonly denoted as $\nabla e(X)\equiv\nabla_X e$ and in the case of $\Gamma(E)=\mathcal{T}(M)$ is exactly how Lee introduces a linear connection.