• *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.

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*Addendum:* It seems Einstein was also not quite consistent with respect to the minus sign; in a <A HREF="https://www.biodiversitylibrary.org/item/92711#page/1140/mode/1up">1914 paper</A> he defined the $\Gamma$ without the minus sign:

<IMG SRC="https://i.sstatic.net/MYf34.png" WIDTH="400"/>

In an interesting discussion on <A HREF="https://hsm.stackexchange.com/questions/7974/notation-for-christoffel-symbols">HSM</A> it is suggested Einstein chose the symbol $\Gamma$ to refer to the first letter of "Gravitation".