From Schwartz's QFT textbook, under Ch.18, Mass renormalization,

Schwartz introduces new LSZ formula with renormalized Green's function.

He states that new LSZ formula for QED, with pole mass $m_p$, is

$$<f|S|i> = (\slash{p_f} - m_p)\dots (\slash{p_i} - m_p) <\psi^R\dots\psi^R>_{amputated}$$ (sorry I don't know how to type Dirac slash.)

where amputated diagrams signify external lines chopped off until they begin interacting with the other fields.

But don't we get amputated diagram after we apply the $(\slash{p_j}-m_p)$ onto the Green's function to project it to S-matrix? I don't understand why external lines are already 'amputated' even before applying $(\slash{p_j}-m_p)$.

Also, another side question, is pole mass defined to be physical mass by definition? I don't understand what forces pole mass to be the actual observable physical mass.