I have struggled for quite some time with the covariant derivative of vector fields. It seems that my confusion arises as a result of different definitions in various textbooks.
Some authors introduce the covariant derivative, $(\nabla _{\vec{v}} \vec{u})_p$, at a point $p$ as a map from a vector field $\vec{u}$ in the neighbourhood of $p$ and a tangent vector $\vec{v}$ at $p$ that is: linear in $v$; additive in $u$; obeys the product rule; commutes with contractions; and reduces to partials derivatives on scalars. So,
$$ \nabla _{\nu} u^\mu = \partial_\nu u^\mu + \Gamma^\mu _{\nu\lambda} u^\lambda $$
where
$$ \nabla _{\vec{e}_\nu} \equiv \nabla _{\nu} \quad \text{and} \quad \nabla _{\nu} \vec{e}_\mu \equiv \Gamma^\lambda _{\mu\nu} \vec{e}_\lambda $$
I also understand that if we demand metric compatibility and insist that the connection is torsion-free then we have done all that is required to uniquely specify a connection called the Levi-Civita Connection. I am comfortable with all of the above - there is nothing to clarify so far.
However, I have also come across many other texts that state that $\partial _{\nu} \vec{e}_\mu \equiv \Gamma^\lambda _{\mu\nu} \vec{e}_\lambda$ and it is usually accompanied by an assertion that the Christoffel symbols are symmetric in their lower two indices because partials commute and $\vec{e}_\mu = \partial_\mu$. The symbols are introduced something like this:
$$ \partial _{\nu} \vec{v} = \partial _{\nu} (v^\mu \vec{e}_\mu) = \partial _{\nu} (v^\mu) \vec{e}_\mu + v^\mu \partial _{\nu} ( \vec{e}_\mu) = (\partial _{\nu} v^\mu + v^\lambda \Gamma^\mu _{\lambda\nu}) \vec{e}_\mu $$
In what way is this correct? To my mind, it feels wrong because a tangent vector should be a linear derivation but $\partial _{\nu} \vec{e}_\mu = \frac{\partial^2}{\partial x^\mu \partial x^\nu}$ is clearly not a Liebnizian operator. Is there any sense in which the partial derivative of a basis vector is legitimately a vector? (I am familiar with Lie brackets but that is different because the second derivatives cancel out). If it's not correct, why do so many texts insist on introducing the symbols in this way?
Wolfram seems to suggest that there are two definitions of the Christoffel symbols floating around but I really don't understand in what circumstances it is valid to state that $\partial _{\nu} \vec{e}_\mu \equiv \Gamma^\lambda _{\mu\nu} \vec{e}_\lambda$ is a vector (although the symmetry in $\mu$ and $\nu$ obviously follows if that is true).
Unfortunately, there are two different definitions of the Christoffel symbol of the second kind. .. The symmetry of definition (6) means that $\Gamma^k _{ij}=\Gamma^k _{ji}$