GR with Torsion: Definition of contorsion I start doing some computations in manifolds with non vanishing torsion and things are getting a bit confused, basically because of notations and definitions. I understand that in presence of non vanishing torsion, one has two fundamental objects:
\begin{align}
\text{The metric} \quad g_{\mu\nu} & & \text{The torsion tensor} \quad T{^\rho}_{\mu\nu} 
\end{align}
The Torsion tensor is defined as 
\begin{equation}
T{^\rho}_{\mu\nu} \equiv \tilde{\Gamma}{^\rho}_{\mu\nu}-\tilde{\Gamma}{^\rho}_{\nu\mu} = 2\tilde{\Gamma}{^\rho}_{[\mu\nu]}
\end{equation}
where $\tilde{\Gamma}{^\rho}_{\mu\nu}$ is the affine connection so that the covariant derivatives $\nabla_{\mu}$ of any vector $A^{\rho}$ and any form $a_{\nu}$ are respectively
\begin{align}
\nabla_{\mu}A^{\rho} = \partial_{\mu}A^{\rho} +\tilde{\Gamma}{^\rho}_{\mu\nu}A^{\nu} & & \nabla_{\mu}a^{\nu} = \partial_{\mu}a^{\nu} -\tilde{\Gamma}{^\rho}_{\mu\nu}a^{\rho} & & 
\end{align}
In Shapiro's article (2001) it says (page 8, equations (2.8) and 2.9)):

The metricity condition
  enables one to express the connection through the metric and torsion in a unique way as
  \begin{equation}
\tilde{\Gamma}{^\rho}_{\mu\nu} = \Gamma{^\rho}_{\mu\nu}+K{^\rho}_{\mu\nu}
\end{equation}
  where $K{^\rho}_{\mu\nu}$ is the contorsion tensor define as
  \begin{equation}
K{^\rho}_{\mu\nu} = \frac{1}{2}\left(T{^\rho}_{\mu\nu}-T{_\mu}{^\rho}{_\nu}-T{_\nu}{^\rho}{_\mu}\right) \tag{1}
\end{equation}

I tried to derive this last formula, but I find a different result. Here's my derivation:
\begin{align}
\nabla_{\mu}g_{\rho\nu} & = \partial_{\mu}g_{\rho\nu}-\tilde{\Gamma}{^\sigma}_{\mu\rho}g_{\sigma\nu}-\tilde{\Gamma}{^\sigma}_{\mu\nu}g_{\rho\sigma} = 0 \\
\nabla_{\nu}g_{\rho\mu} & = \partial_{\nu}g_{\rho\mu}-\tilde{\Gamma}{^\sigma}_{\nu\rho}g_{\sigma\mu}-\tilde{\Gamma}{^\sigma}_{\nu\mu}g_{\rho\sigma} = 0 \\
-\nabla_{\rho}g_{\mu\nu} & = -\partial_{\rho}g_{\mu\nu}+\tilde{\Gamma}{^\sigma}_{\rho\mu}g_{\sigma\nu}+\tilde{\Gamma}{^\sigma}_{\rho\nu}g_{\mu\sigma} = 0 
\end{align}
Summing up these three equations one gets
\begin{align}
{\color{blue}{\Big(\partial_{\mu}g_{\rho\nu}+\partial_{\nu}g_{\rho\mu}-\partial_{\rho}g_{\mu\nu}\Big)}} +g_{\nu\sigma}\Big(\tilde{\Gamma}{^\sigma}_{\rho\mu}-\tilde{\Gamma}{^\sigma}_{\mu\rho}\Big)& +g_{\mu\sigma}\Big(\tilde{\Gamma}{^\sigma}_{\rho\nu}-\tilde{\Gamma}{^\sigma}_{\nu\rho}\Big)+\\
& {\color{red}{-g_{\rho\sigma}\Big(\tilde{\Gamma}{^\sigma}_{\mu\nu}+\tilde{\Gamma}{^\sigma}_{\nu\mu}\Big)}} = 0 \tag{2}
\end{align}
Using the fact that $\tilde{\Gamma}{^\sigma}_{\nu\mu} = \tilde{\Gamma}{^\sigma}_{\mu\nu}+T{^\sigma}_{\nu\mu}$, and introducing the Christoffel symbols 
\begin{equation}
\Gamma{^\rho}_{\mu\nu} = \frac{1}{2}g^{\rho\sigma}\left(\partial_{\mu}g_{\sigma\nu}+\partial_{\nu}g_{\mu\sigma}-\partial_{\sigma}g_{\mu\nu}\right)
\end{equation}
We can rewrite (2) as
\begin{equation}
{\color{red}{2g_{\rho\sigma}\tilde{\Gamma}{^\sigma}_{\mu\nu}}} = {\color{blue}{2g_{\rho\sigma}\Gamma{^\sigma}_{\mu\nu}}}{\color{red}{-g_{\rho\sigma}T{^\sigma}_{\nu\mu}}}+g_{\nu\sigma}T{^\sigma}_{\rho\mu}+g_{\mu\sigma}T{^\sigma}_{\rho\nu}
\end{equation}
Hence we get
\begin{align}
\tilde{\Gamma}{^\rho}_{\mu\nu} & = \Gamma{^\rho}_{\mu\nu} -\frac{1}{2}\Big(T{^\rho}_{\nu\mu}-T{_\mu}{^\rho}{_\nu}-T{_\nu}{^\rho}{_\mu}\Big) = \\
& =  \Gamma{^\rho}_{\mu\nu} +\frac{1}{2}\Big(T{^\rho}_{\mu\nu}+T{_\mu}{^\rho}{_\nu}+T{_\nu}{^\rho}{_\mu}\Big) =  \Gamma{^\rho}_{\mu\nu}+K{^\rho}_{\mu\nu} \tag{3}
\end{align} 
Obviously the contorsion tensors in (1) and (3) do not agree. Actually the definition of $K{^\rho}_{\mu\nu}$ that I got can be written as
\begin{equation}
K{^\rho}_{\mu\nu} = \frac{1}{2}\Big(T{^\rho}_{\mu\nu}-T{_{\mu\nu}}{^\rho}+T{_\nu}{^\rho}{_\mu}\Big)
\end{equation}
which is the definition that I found in some others references. What is going on? Ultimately what should I take as definition of the contorsion?

Edit The main problem here is that (if I did things correctly) using the conventions of Shapiro (2001) for the torsion and the contorsion tensors, I eventually get a result different from his.
 A: I decided to answer my own question because, hopefully, I found at least some coherent results.

In Shapiro's(2001) something doesn't work: I say this because according to his definition
\begin{align}
\tilde{\Gamma}{^\rho}_{\mu\nu} = \Gamma{^\rho}_{\mu\nu} + K{^\rho}_{\mu\nu} & & K{^\rho}_{\mu\nu} = \frac{1}{2}\left(T{^\rho}_{\mu\nu}-T{_\mu}{^\rho}{_\nu}-T{_\nu}{^\rho}{_\mu}\right)
\end{align}
Of course the affine connection must satifies the metricity condition $\nabla_\rho g_{\mu\nu}=0$. Therefore we get the equation
\begin{equation}
 \nabla_\sigma g_{\mu\nu}=\partial_\sigma g_{\mu\nu}-\Gamma{^\rho}_{\sigma\mu}g_{\rho\nu}-\Gamma{^\rho}_{\sigma\nu}g_{\mu\rho}-K{^\rho}_{\sigma\mu}g_{\rho\nu}-K{^\rho}_{\sigma\nu}g_{\mu\rho} 
\end{equation}
Substituting the expression of the Christoffel symbols, the first three terms of the above equations cancel and one is left with (pulling all indices down for simplicity)
\begin{equation}
-K_{\nu\sigma\mu}-K_{\mu\sigma\nu} = -\left(T_{\nu\sigma\mu}-T_{\sigma\nu\mu}-T_{\mu\nu\sigma}\right)\neq 0
\end{equation}
Hence the metricity condition is not satisfied; hence, either I did something wrong in these computations, either something is not well define in the paper.

Then I found two (of course equivalent) order-of-indices conventions in which torsion and contorsion are often introduced:
First way: Having the metric $g_{\mu\nu}$ and the affine connection $\tilde{\Gamma}{^\rho}_{\mu\nu}$, one define the torsion as
\begin{equation}
T{^\rho}_ {{\color{green}{\mu\nu}}} \equiv 2\tilde{\Gamma}{^\rho}_{[\mu\nu]} = \tilde{\Gamma}{^\rho}_{\mu\nu} -\tilde{\Gamma}{^\rho}_{\nu\mu}
\end{equation}
I decide to write in green indices that in a tensor are antisymmetric. Hence in this first convention the torsion tensor is antisymmetric in the last two indices. One then can again compute 
\begin{align}
\nabla_{\mu}g_{\rho\nu} & = \partial_{\mu}g_{\rho\nu}-\tilde{\Gamma}{^\sigma}_{\mu\rho}g_{\sigma\nu}-\tilde{\Gamma}{^\sigma}_{\mu\nu}g_{\rho\sigma} = 0 \tag{1.a}\\
\nabla_{\nu}g_{\rho\mu} & = \partial_{\nu}g_{\rho\mu}-\tilde{\Gamma}{^\sigma}_{\nu\rho}g_{\sigma\mu}-\tilde{\Gamma}{^\sigma}_{\nu\mu}g_{\rho\sigma} = 0 \tag{1.b}\\
-\nabla_{\rho}g_{\mu\nu} & = -\partial_{\rho}g_{\mu\nu}+\tilde{\Gamma}{^\sigma}_{\rho\mu}g_{\sigma\nu}+\tilde{\Gamma}{^\sigma}_{\rho\nu}g_{\mu\sigma} = 0 \tag{1.c}
\end{align}
Summing up these three equations one gets
\begin{equation}
\tilde{\Gamma}{^\rho}_{\mu\nu} = \Gamma{^\rho}_{\mu\nu}+\frac{1}{2}\left(T{^\rho}_{{\color{green}{\mu\nu}}}-T_{\mu{\color{green}{\nu}}}{^{\color{green}{\rho}}}+T_{\nu}{^{\color{green}{\rho}}}_{{\color{green}{\mu}}}\right)
\end{equation}
It's easy to verify that the combination in round brackets is antisymmetric in the exchange of $\nu$ and $\rho$. Then one can define the contorsion to be 
\begin{equation}
K_{\mu{\color{green}{\nu}}}{^{\color{green}{\rho}}} \equiv \frac{1}{2}\left(T{^\rho}_{{\color{green}{\mu\nu}}}-T_{\mu{\color{green}{\nu}}}{^{\color{green}{\rho}}}+T_{\nu}{^{\color{green}{\rho}}}_{{\color{green}{\mu}}}\right)
\end{equation}
Second way: Having again the metric $g_{\mu\nu}$ and the affine connection $\tilde{\Gamma}{^\rho}_{\mu\nu}$, one can define the torsion switching the order of the indices as follows
\begin{equation}
T_ {{\color{green}{\mu\nu}}}{^\rho} \equiv 2\tilde{\Gamma}{^\rho}_{[\mu\nu]} = \tilde{\Gamma}{^\rho}_{\mu\nu} -\tilde{\Gamma}{^\rho}_{\nu\mu}
\end{equation}
Therefore in this second convention the torsion is antisymmetric in the first two indices. Again one can write the three equations (1.a) (1.b) and (1.c); their sum gives 
\begin{equation}
\tilde{\Gamma}{^\rho}_{\mu\nu} = \Gamma{^\rho}_{\mu\nu}+\frac{1}{2}\left(T_{{\color{green}{\mu\nu}}}{^\rho}-T_{{\color{green}{\nu}}}{^{\color{green}{\rho}}}_{\mu}+T{^{\color{green}{\rho}}}_{{\color{green}{\mu}}\nu}\right)
\end{equation}
Again the quantity in round brackets is antisymmetric in $\nu$ and $\rho$, and the contorsion can be define as
\begin{equation}
K_{\mu{\color{green}{\nu}}}{^{\color{green}{\rho}}} \equiv \frac{1}{2}\left(T_{{\color{green}{\mu\nu}}}{^\rho}-T_{{\color{green}{\nu}}}{^{\color{green}{\rho}}}_{\mu}+T{^{\color{green}{\rho}}}_{{\color{green}{\mu}}\nu}\right)
\end{equation}
Of course one can change the order of the indices also in the contorsion tensor. For istance in the second way (or analogously for the first case) one can say 
\begin{equation}
K{^{\color{green}{\rho}}}_{{\color{green}{\nu}}\mu} \equiv \frac{1}{2}\left(T_{{\color{green}{\mu\nu}}}{^\rho}-T_{{\color{green}{\nu}}}{^{\color{green}{\rho}}}_{\mu}+T{^{\color{green}{\rho}}}_{{\color{green}{\mu}}\nu}\right) 
\end{equation}
This of course doesn't change the order of indices on the combination on the right hand side. 

Finally, following the answer of @Saksith Jaksri in Torsion tensor: definition, I stress that for each paper one should be careful on how the "covariant derivative is define", namely
\begin{align}
\nabla_{{\color{red}{\mu}}}A^{\rho} & = \partial_{\mu}A^{\nu} +\tilde{\Gamma}{^\rho}_{{\color{red}{\mu}}\nu}A^{\nu} \tag{A}\\
\nabla_{{\color{red}{\mu}}}A^{\rho} & = \partial_{\mu}A^{\nu} +\tilde{\Gamma}{^\rho}_{\nu{\color{red}{\mu}}}A^{\nu} \tag{B}
\end{align}
In my answer I used the convention (A). Of course in case (B) is used things slightly change. So when torsion start to play a role, one has to keep track of all order-of-indices convention of each author.

final edit: Actually it seems to me that if one uses convention (B) in Shapiro(2001), everything seems to work, also for some other equations written later in the paper, even though in equation (2.1) (page 6) it clearly introduce the covariant derivative following (A). However I have to say that I didn't read the entire paper, so I can't be completely sure of this possible solution.
A: Here are expressions for the various conventions after slogging through the calculations:
If the connection coefficients are defined such that $\nabla_{a}w^{b}=\partial_{a}w^{b}+\Gamma^{b}{}_{ca}w^{c}$, then the usual definition of the torsion as $\vec{T}\left(v,w\right)=\nabla_{v}w-\nabla_{w}v$ corresponds to the tensor $T^{c}{}_{ab}=\Gamma^{c}{}_{ba}-\Gamma^{c}{}_{ab}$, which results in the contorsion tensor
$$K_{abc} = \frac{1}{2}\left(T_{bac}+T_{cab}-T_{abc}\right).$$
For whatever reason, many papers seem to stick with connection coefficients $\nabla_{a}w^{b}=\partial_{a}w^{b}+\Gamma^{b}{}_{ca}w^{c}$ but instead define the torsion tensor with a reversed sign $T^{c}{}_{ab}=\Gamma^{c}{}_{ab}-\Gamma^{c}{}_{ba}$; this then flips the sign of the contorsion tensor, which can then be written
$$\begin{aligned}K_{abc} & = -\frac{1}{2}\left(T_{bac}+T_{cab}-T_{abc}\right)\\  & =\frac{1}{2}\left(T_{abc}-T_{bac}-T_{cab}\right). \end{aligned}$$
This is the approach taken on the Wikipedia page at the time of this writing, and is also the expression in Shapiro, which as the OP noted is inconsistent with connection coefficients defined such that $\nabla_{a}w^{b}=\partial_{a}w^{b}+\Gamma^{b}{}_{ac}w^{c}$; instead, this connection coefficient convention, when paired with the torsion tensor definition $T^{c}{}_{ab}=\Gamma^{c}{}_{ab}-\Gamma^{c}{}_{ba}$, which is in this case consistent with $\vec{T}\left(v,w\right)=\nabla_{v}w-\nabla_{w}v$, results in a contorsion tensor
$$K_{abc} = \frac{1}{2}\left(T_{abc}+T_{cab}+T_{bac}\right).$$
Although I didn’t come across this in the literature, if one was to instead define torsion with the opposite sign, this would again flip the sign of the contorsion tensor.
These expressions also remain valid in an anholonomic frame, where
$$\vec{T}\left(v,w\right)=\nabla_{v}w-\nabla_{w}v-\left[v,w\right]$$
and
$$\begin{aligned}2\Gamma^{c}{}_{ba}=g^{cd}( & \partial_{a}g_{bd}+\partial_{b}g_{da}-\partial_{d}g_{ab}\\ & -g_{af}[e_{b},e_{d}]^{f}+g_{bf}[e_{d},e_{a}]^{f}+g_{df}[e_{a},e_{b}]^{f}). \end{aligned}$$ 
