(Coleman's lecture note) scattering in QFT

I am currently reading Coleman's lecture note on QFT.(https://arxiv.org/abs/1110.5013) I have several questions regarding the scattering theory. Let $$\phi$$ be a real scalar field, and consider the interaction Hamiltonian $$H_I(x)=g\phi(x)\rho(\vec{x})f(t)$$, where $$f$$ represents the adiabatic turning on/off function.

The following is pp.91-92 of the lecture note:

Questions:

1. What is the difference between the vacuum with respect to $$H_0$$, and the vacuum with respect to $$H_0 + H_I$$? In the lecture note, these two vacuums are denoted by different notations $$|0\rangle$$ and $$|0\rangle_P$$. What is their difference?

2. Somewhat ad hoc counterterm $$a=E_0$$ was added to $$H_I$$ to fix the divergent phase in $$\langle0|S|0\rangle$$. Adding $$a$$ will fix the divergent phase, but I think it does not necessarily means that $$\langle0|S|0\rangle =1$$. (As described in the lecture note, the correct expression should be $$\langle0|S|0\rangle=e^{-i(\gamma_- +\gamma_+)}$$.

1. The states $$|0\rangle~\in~{\cal H}_0 \quad\text{and}\quad |0\rangle_P~\in~{\cal H}$$ are the ground states for the Hamiltonians $$\hat{H}_0$$ and $$\hat{H}$$, respectively. More explicitly, using Coleman's notation p. 72-77, we have in the Schrödinger picture
$$|0 (t)\rangle_S~=~|0 (t)\rangle_S^{\rm in} ~=~e^{-i\gamma_-}|0 (t)\rangle_{P,S}\quad\text{for}\quad t\lesssim -\frac{T}{2}$$ and
$$|0 (t)\rangle_S~=~|0 (t)\rangle_S^{\rm out} ~=~e^{i\gamma_+}|0 (t)\rangle_{P,S}\quad\text{for}\quad t\gtrsim \frac{T}{2}.$$
2. The phase $$e^{-i(\gamma_- +\gamma_+)}$$ is apparently absorbed via the $$O(T/\Delta)$$ counterterm.