Generally the Christoffel symbol of the first kind is defined as
$$\Gamma_{\lambda\mu\nu}=\frac12\,(\partial_\nu g_{\lambda\mu}+\partial_\mu g_{\lambda\nu}-\partial_\lambda g_{\mu\nu}) \tag{1}$$
and the Christoffel symbol of the second kind is defined as
$$\Gamma^\rho{}_{\mu\nu}=g^{\rho\lambda}\Gamma_{\lambda\mu\nu}=\frac12\,g^{\rho\lambda}\,(\partial_\nu g_{\lambda\mu}+\partial_\mu g_{\lambda\nu}-\partial_\lambda g_{\mu\nu}) \tag{2}.$$
Is the following definition of $\Gamma_{\mu\nu}{}^\rho$ correct or should it be something else?
\begin{align} \Gamma_{\mu\nu}{}^\rho=g^{\rho\lambda}\Gamma_{\mu\nu\lambda}&=\frac12\,g^{\rho\lambda}\,(\partial_\lambda g_{\mu\nu}+\partial_\nu g_{\mu\lambda}-\partial_\mu g_{\nu\lambda}) \tag{3} \\ &=g^{\rho\lambda}\,\partial_\nu g_{\lambda\mu}-\Gamma^\rho{}_{\mu\nu}. \end{align}