 Solution

Write $g=\det (g_{ab})$. The Jacobi formula gives $\delta g=g{g}^{ab}\delta g_{ab}$. Since a Lorentzian metric of signature $(+---)$ has negative determinant,The variation of metric volume density therefore has the opposite sign when expressed using the inverse metric tensor: $\delta \sqrt{-g}=-\sqrt{-g}{g}_{ab}\delta g^{ab}/2$. This follows immediately from the variation of inverse metric and explains the volume term in the field-equation derivation.