• *Why Einstein introduced a minus sign in the definition of the second Christoffel symbol $\Gamma^\sigma_{\mu\nu}$:*    
He writes just below equation (45) in Ref. 1:

<IMG SRC="https://i.sstatic.net/lsetA.png" WIDTH="400"/>

So he wanted to identify the $\Gamma^\sigma_{\mu\nu}$ with the components of the gravitational field, and for that identification the minus sign is needed.

• *Why was the minus sign dropped?*    
It was understood that the identification of the Christoffel symbol with the gravitational field is mistaken: you can have a nonzero $\Gamma^\sigma_{\mu\nu}$ and zero gravitational field, all you have to do is to introduce curved coordinate systems in flat space. And conversely, the Christoffel symbol can vanish along a geodesic even if the gravitational field is nonzero.



  [1]: https://i.sstatic.net/lsetA.png