Recently I understood that the energy momentum tensor can be calculated by:
\begin{equation} T_{\mu \nu}=\frac{2}{\sqrt{-g}}\frac{\delta S_m}{\delta g^{\mu \nu}}.\tag{1} \end{equation}
So consider the action
\begin{equation} S_m=\int d^4x\sqrt{-g}\times \frac{1}{2}(g^{\mu \nu}\partial_{\mu}\phi \partial_{\nu}\phi-m^2\phi^2).\tag{2} \end{equation}
Here I'm using the $(+,-,-,-)$ Minkowski sign convention. Varying this action with respect to $g^{\mu\nu}$, I obtained (the factor of $\frac{1}{2}$ dropped temporarily):
\begin{equation} \begin{split} &\quad \frac{\delta \sqrt{-g}}{\delta g^{\mu \nu}}(g^{\mu \nu}\partial_{\mu}\phi \partial_{\nu}\phi-m^2\phi^2) + \sqrt{-g} \frac{\delta}{\delta g^{\mu \nu}}(g^{\alpha \beta}\partial_{\alpha}\phi \partial_{\beta}\phi)\\ &=-\frac{\sqrt{-g}}{2}g_{\mu\nu}(g^{\mu \nu}\partial_{\mu}\phi \partial_{\nu}\phi-m^2\phi^2) + \sqrt{-g}\left(\frac{1}{2}\delta_{\mu}^{\alpha}\delta_{\nu}^{\beta}+\frac{1}{2}\delta_{\nu}^{\alpha}\delta_{\mu}^{\beta} \right)\partial_{\alpha}\phi \partial_{\beta}\phi \\ &=\sqrt{-g}\left( \partial_{\mu}\phi \partial_{\nu}\phi - \frac{1}{2} g_{\mu \nu}(g^{\alpha \beta}\partial_{\alpha}\phi \partial_{\beta}\phi-m^2\phi^2)\right) \end{split}\tag{3} \end{equation}
which yields: $$ T_{\mu \nu} = \partial_{\mu}\phi \partial_{\nu}\phi - \frac{1}{2} g_{\mu \nu}(g^{\alpha \beta}\partial_{\alpha}\phi \partial_{\beta}\phi-m^2\phi^2)\tag{4} $$ and is consistent with the one given by Noether theorem.
But if I write the action as
\begin{equation} S_m=\int d^4x\sqrt{-g}\times \frac{1}{2}(g_{\mu \nu}\partial^{\mu}\phi \partial^{\nu}\phi-m^2\phi^2)\tag{5} \end{equation}
then the identity: $$ \delta g_{\mu \nu} = - g_{\mu \alpha} g_{\nu \beta} \delta g^{\alpha \beta}\tag{6} $$ since to be introducing an additional minus sign so that the energy momentum tensor is $$ T_{\mu \nu} = -\partial_{\mu}\phi \partial_{\nu}\phi - \frac{1}{2} g_{\mu \nu}(g^{\alpha \beta}\partial_{\alpha}\phi \partial_{\beta}\phi-m^2\phi^2).\tag{7} $$
Is it that $$g_{\mu \nu}\partial^{\mu}\phi \partial^{\nu}\phi \neq g^{\mu \nu}\partial_{\mu}\phi \partial_{\nu}\phi\tag{8}$$ in GR or have I done something wrong? (I have never studied GR before, so much appreciate if anyone find other stuff I've done wrong)
