What's the meaning of the Feynman propagator for the driven quantum harmonic oscillator?

Consider a quantum harmonic oscillator that is driven for a finite time by a force $J(t)$, and work entirely in Heisenberg picture. Then we may define the 'in' and 'out' vacua $$|0_{\text{in}} \rangle, \quad |0_{\text{out}} \rangle$$ to be the ground states of the Hamiltonian at early and late times. In Schrodinger picture, the 'in' vacuum corresponds to a state in the usual QHO ground state before the driving starts, while the 'out' vacuum corresponds to a state that ends up in that state when the driving ends.

In Mukhanov and Winitzki's book the retarded Green's function is defined as a matrix element between 'in' states, $$\langle 0_{\text{in}} | \hat{q}(t) | 0_{\text{in}}\rangle = \int J(t') G_{\text{ret}}(t, t') \, dt', \quad G_{\text{ret}}(t, t') = \frac{\sin \omega(t - t')}{\omega} \theta(t - t').$$ This makes perfect sense thinking semiclassically, as $\langle \hat{q}(t) \rangle$ is just the average position of the particle, given that it was at rest in the far past; that's basically the definition of what a retarded propagator is. Similarly, one can define the advanced propagator using $|0_{\text{out}}\rangle$.

Finally, Mukhanov and Winitzki define the Feynman propagator by $$\langle 0_{\text{out}} | \hat{q}(t) | 0_{\text{in}}\rangle \propto \int J(t') G_{F}(t, t') \, dt'.$$ Now, I've been searching for an intuitive understanding of the Feynman propagator for years. Typical explanations in quantum field theory speak of "negative energy solutions" and antiparticles (e.g. here) which I've always been confused about, since they don't exist in ordinary quantum mechanics (as I asked here). But above we have a Feynman propagator for an exceptionally simple non-QFT system! So if there's an intuitive explanation at all it'll be right here, but I can't quite see what the matrix element means physically.

I have two questions: first, how is this equivalent to the usual definition of the Feynman propagator, involving a particular contour choice? Second, are there intuitive words one can drape around this definition? Does it provide any additional physical insight?