At several points throughout Weinberg QFT Volume I, Weinberg claims that the sum of all diagrams which have in states $\alpha$ and out states $\beta$ and one off-shell photon at position $x$ is given by \begin{equation} \langle \beta|\hat{J}^{\mu}(x)|\alpha\rangle \tag{1} \end{equation} where $J^{\mu}(x)=\frac{\delta L_{QED}}{\delta A^{\mu}(x)}$. For instance he says this at Eq's(10.4.19),(10.5.1) and (13.6.1). However, I am under the impression that the amplitude for single photon emission is rather: \begin{equation} \langle \beta|\hat{A}^{\mu}(x)|\alpha\rangle \tag{2} \end{equation} Below Eq.(10.5.1) he says Eq(1) is the case in theories such as QED where the interaction is linear in $A^{\mu}(x)$. I do not understand this comment as Eq(2) is the amplitude for single photon emission?
My second question: In QCD, does only Eq(2) work for describing single gluon emission, supposing that we insert the gluon operator in place of the photon field operator?