Someone could me explain how Einstein goes from this:
\begin{align*} \frac{\partial g^{\sigma\beta} Γ^\alpha_{µ\beta}}{\partial x^\alpha} &=−\Omega\left[(t^\sigma _\mu+T^\sigma_\mu)−\frac12\delta^\sigma_\mu (t+T)\right] \\ (−g)^{1/2} &=1 \end{align*}
to this:
\begin{align*} \frac{\partial\Gamma^\alpha_{\mu\nu}}{\partial x^\alpha}+\Gamma^\alpha_{\mu\beta} \Gamma^\beta_{\nu\alpha}&=−\Omega\left[T_{\mu\nu} −\frac12g_{\mu\nu}T\right]\\ (−g)^{1/2} &=1 \end{align*}
I studied from the Einstein's book "Zur Elektrodynamik bewegter Körper [1905] - Die Grundlage der allgemeinen Relativitätstheorie [1916]" (translated in Italian language) and there is no connection, as it seems to me, between the two equations(there are affinities) .
As you can see in this english translation of the second article you mentioned, "Die Grundlage der allgemeinen Relativitätstheorie", eq (50):
$\varkappa t_{\sigma}^{\alpha}=\frac{1}{2}\delta_{\sigma}^{\alpha}g^{\mu\nu}\Gamma_{\mu\beta}^{\lambda}\Gamma_{\nu\lambda}^{\beta}-g^{\mu\nu}\Gamma_{\mu\beta}^{\alpha}\Gamma_{\nu\sigma}^{\beta}$
Using metric compatibility of the Levi-Civita connection ($\nabla g=0$), we get:
$\partial_{\alpha}g^{\mu\nu}=\Gamma^{\mu}_{i \alpha}g^{i\nu}+\Gamma^{\nu}_{ \alpha i}g^{\mu i}$
Expanding $t$ and using this relation:
$\partial_{\alpha}\left(g^{\sigma\beta}\Gamma^{\alpha}_{\mu\beta}\right)+ \varkappa\left(t^{\sigma}_{\mu}-\frac{1}{2}\delta^{\sigma}_{\mu}t\right)= -\varkappa\left[\left(T_{\mu}^{\sigma}-\frac{1}{2}\delta^{\sigma}_{\mu}T\right)\right]$
$\partial_{\alpha}\left(g^{\sigma\beta}\right)\Gamma^{\alpha}_{\mu\beta}+g^{\sigma\beta}\partial_{\alpha}\left(\Gamma^{\alpha}_{\mu\beta}\right)+ \frac{1}{2}\delta^{\sigma}_{\mu}g^{i\nu}\Gamma^{\lambda}_{i\beta}\Gamma^{\beta}_{\nu\lambda}-g^{i\nu}\Gamma^{\sigma}_{i\beta}\Gamma^{\beta}_{\nu\mu}-\frac{1}{2}\delta^{\sigma}_{\mu}g^{i\nu}\Gamma^{\lambda}_{i\beta}\Gamma^{\beta}_{\nu\lambda}=-\varkappa\left[\left(T_{\mu}^{\sigma}-\frac{1}{2}\delta^{\sigma}_{\mu}T\right)\right]$
$\left[\Gamma^{\sigma}_{i \alpha}g^{i\beta}+\Gamma^{\beta}_{ \alpha i}g^{\sigma i}\right]\Gamma^{\alpha}_{\mu\beta}+g^{\sigma \beta}\partial_{\alpha}\left(\Gamma^{\alpha}_{\mu\beta}\right)-g^{i\nu}\Gamma^{\sigma}_{i\beta}\Gamma^{\beta}_{\nu\mu}=-\varkappa\left[\left(T_{\mu}^{\sigma}-\frac{1}{2}\delta^{\sigma}_{\mu}T\right)\right]$
$g^{\sigma i}\Gamma^{\beta}_{\alpha i}\Gamma^{\alpha}_{\mu\beta}+g^{\sigma\beta}\partial_{\alpha}\left(\Gamma^{\alpha}_{\mu\beta}\right)=-\varkappa\left[\left(T_{\mu}^{\sigma}-\frac{1}{2}\delta^{\sigma}_{\mu}T\right)\right]$
Multiplying both sides for $g_{\sigma \tau}$:
$\delta^{i}_{\tau}\Gamma^{\beta}_{\alpha i}\Gamma^{\alpha}_{\mu\beta}+\delta^{\beta}_{\tau}\partial_{\alpha}\left(\Gamma^{\alpha}_{\mu\beta}\right)=-g_{\sigma \tau}\varkappa\left[\left(T_{\mu}^{\sigma}-\frac{1}{2}\delta^{\sigma}_{\mu}T\right)\right]$
$\Gamma^{\beta}_{\alpha \tau}\Gamma^{\alpha}_{\mu\beta}+\partial_{\alpha}\left(\Gamma^{\alpha}_{\mu\tau}\right)=-\varkappa\left[\left(T_{\mu \tau}-\frac{1}{2}g_{\mu\tau}T\right)\right]$
Renaming $\tau=\nu$:
$\Gamma^{\beta}_{\alpha \nu}\Gamma^{\alpha}_{\mu\beta}+\partial_{\alpha}\left(\Gamma^{\alpha}_{\mu\nu}\right)=-\varkappa\left[\left(T_{\mu \nu}-\frac{1}{2}g_{\mu\nu}T\right)\right]$
NOTE: This equation is yet not fully covariant, since the left member is not a tensor: it is, instead, a tensor density. We can generalize it noting that, using (44) in Einstein's work:
$\begin{cases} B_{\mu\nu} & =R_{\mu\nu}+S_{\mu\nu}\\ \\R_{\mu\nu} & =-\frac{\partial}{\partial x_{\alpha}}\left\{ {\mu\nu\atop \alpha}\right\} +\left\{ {\mu\alpha\atop \beta}\right\} \left\{ {\nu\beta\atop \alpha}\right\} \\ \\S_{\mu\nu} & =\frac{\partial\lg\sqrt{-g}}{\partial x_{\mu}\partial x_{\nu}}-\left\{ {\mu\nu\atop \alpha}\right\} \frac{\partial\lg\sqrt{-g}}{\partial x_{\alpha}}\end{cases}$
In the particular case in which $\sqrt{-g}=1$, we get $B_{\mu\nu}=R_{\mu\nu}$.
Thus, the equation becomes: $B_{\mu\nu}=-\varkappa\left[\left(T_{\mu \nu}-\frac{1}{2}g_{\mu\nu}T\right)\right]$
By contraction, we note that $B=B^{\mu}_{\mu}=-\varkappa\left[T-\frac{d}{2}T\right]$
In four dimensions, we have: $B=\varkappa T$
So we can write
$B_{\mu\nu}-\frac{B}{2}g_{\mu\nu}=-\varkappa T_{\mu\nu}$
These are the famous Einstein field equation (with a minus sign in front of $T$ compared to the canonical version since Einstein has defined ricci curvature tensor with the signs reversed compared to the canonical definition)