Let $\pi: E \rightarrow M$ be a vector bundle over a manifold $M$, $\mathfrak{X}(M)$ the set of vector fields over $M$, and $\mathcal{E}(M)$ the space of smooth sections of $E$. Then for a given connection $\nabla$, $Y \in \mathcal{E}(M)$, and $X \in \mathfrak{X}(M)$ we call $\nabla_X Y \in \mathcal{E}(M)$ a covariant derivative. If $Y$ is a $(p,q)$ tensor field then so is its covariant derivative $\nabla_X Y$.
Now if $\nabla$ is a linear/affine connection, we may also define the $(p, q+1)$ tensor field $\nabla Y$ as $$\nabla Y(\omega^1, \ldots, \omega^p, V_1, \ldots, V_q, X) = \nabla_XY(\omega^1, \ldots, \omega^p, V_1, \ldots, V_q)$$ which we call the total covariant derivative.
Other than choosing a specific vector field/direction for the covariant derivative, I don't see what is the difference between it and the total covariant derivative. If that is the only difference, what is the point of defining a whole new object? Does $\nabla Y$ geometrically say anything that $\nabla_X Y$ does not?