In Wald's book General Relativity, Wald gives the energy-momentum tensor of the scalar field that leads to the Klein-Gordon equation, $$ \begin{align} T_{\mu\nu} =\phi_{,\mu}\phi_{,\nu}-\frac{1}{2}\eta_{\mu\nu}\left(\phi_{,\sigma}\phi^{,\sigma}+m^2\phi^2\right),\tag{1} \end{align} $$ wherein I used the notation $\phi_{,\mu}:=\partial_\mu\phi$. Eq. (1) is eq. (4.2.20) on p. 63; Wald uses the $(-1,+1,+1,+1)$ convention for the Minkowski metric. The Lagrangian density of the real free and massive scalar field that satisfies the Klein-Gordon equation using said metric convention is given by, $$ \begin{align} \mathcal{L}=-\frac{1}{2}\phi_{,\sigma}\phi^{,\sigma}-\frac{1}{2}m^2\phi^2=-\frac{1}{2}\phi_{,\sigma}\phi^{,\sigma}-V(\phi).\tag{2} \end{align} $$ The former expression can also be found on p. 451 as Eq. (E.1.6). We now write Eq. (1) as, $$ \begin{align} T_{\mu\nu} =\phi_{,\mu}\phi_{,\nu}+\eta_{\mu\nu}\mathcal{L}.\tag{3} \end{align} $$ The energy-momentum tensor with raised indices can be obtained from Eq. (3), $$ \begin{align} T^{\mu\nu} =\eta^{\mu\sigma}\eta^{\nu\rho}T_{\sigma\rho}=\phi^{,\mu}\phi^{,\nu}+\eta^{\mu\sigma}\eta^{\nu\rho}\eta_{\sigma\rho}\mathcal{L} =\eta^{\mu\sigma}\eta^{\nu\rho}T_{\sigma\rho}=\phi^{,\mu}\phi^{,\nu}+\eta^{\mu\nu}\mathcal{L}.\tag{4} \end{align} $$ From now on, we will refer to the energy-momentum tensor of the Klein-Gordon scalar field as represented in Eq. (4).
Okay, so now get to the actual confusion. In the appendix, Wald derives the energy-momentum tensor in eq. (E.1.36) $$ \begin{align} T^{\mu\nu}=\frac{\partial\mathcal{L}}{\partial(\phi_{,\mu})}\phi^{,\nu}-\eta^{\mu\nu}\mathcal{L}\tag{5} \end{align} $$ from Noether's theorem on p. 457 and calls it the canonical energy-momentum tensor. Let us first evaluate, $$ \begin{align} \frac{\partial\mathcal{L}}{\partial(\phi_{,\mu})} =-\frac{1}{2}\eta^{\sigma\rho}\frac{\partial(\phi_{,\sigma}\phi_{,\rho})}{\partial(\phi_{,\mu})} =-\frac{1}{2}\eta^{\sigma\rho}\delta_{\sigma\mu}\phi_{,\sigma}-\frac{1}{2}\eta^{\sigma\rho}\delta_{\sigma\rho}\phi_{,\rho} =-\phi^{,\mu}, \tag{6} \end{align} $$ and insert Eq. (6) into (5). We find, $$ \begin{align} T^{\mu\nu}=-\phi^{,\mu}\phi^{,\nu}-\eta^{\mu\nu}\mathcal{L}. \tag{7} \end{align} $$ It is obvious that Eq. (7) does not equal (4), however, both equations only differ by a sign.
Can it be that the sign difference has something to do with Does metric signature affect the stress energy tensor?
If not, how can one explain the sign difference?
