GR with Torsion: Definition of contorsion

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.