After having studied canonical quantization and feeling (relatively) comfortable with it, I have now been studying the path integral approach. But I don't feel entirely comfortable with.
I have the feeling that the main objective of the path-integral approach is to calculate the Green's function: \begin{equation} G^{(n)}(x_1,\ldots,x_n) = \langle 0 | \mathcal{T} \{\phi(x_1) \cdots \phi(x_n) \} |0\rangle = \left(\frac{1}{i}\right)^n \frac{\delta^n W[J]}{\delta J(x_1) \ldots \delta J(x_n)}\biggr|_{J=0} \end{equation} where, for simplicity, I have considered the neutral scalar field $\phi$ and $\mathcal{T}$ denotes the time-ordering operator. I have troubles understanding the physical meaning of the Green's function.
I understand that for the canonical quantization procedure, i.e. when $\phi$ is a field operator, $G^{(n)}(x_1,\ldots,x_n)$ is the vacuum expectation value. However, if I understand it correctly, in the path-integral approach we consider $\phi$ to be a classical field. I don't understand how rhyme these two different pictures.
Furthermore, for the canonical quantization formalism, we can represent the S-matrix: \begin{equation} S_{fi} = \langle f | S | i \rangle \end{equation} by Feynman diagrams. On the other hand, for the path-integral approach we seem to represent $G^{(n)}(x_1,\ldots,x_n)$ by Feynman diagrams. Do these Feynman diagram for the two different approaches somehow represent the same scattering amplitude?
Basically, I feel like I can't see the forest for the trees, and I am hoping someone can clarify the above described problems.
P.S. we have derived the LSZ reduction formula, and thus I understand that in the canonical quantization formalism we can express the S-matrix elements in terms of $G^{(n)}(x_1,\ldots,x_n)$. However, our lecturer told us that no one really uses the LSZ formula for practical purposes, and thus I don't think this answers my questions.