In the context of Renormalized Pertubation Theory, Peskin & Schroeder says:
The Lagrangian $$ \mathcal{L}=\frac{1}{2} (\partial_\mu\phi_r)^2-\frac{1}{2}m^2\phi_r^2-\frac{\lambda}{4!}\phi_r^4 + \frac{1}{2} \delta_Z(\partial_\mu\phi_r)^2 -\frac{1}{2}\delta_m\phi_r^2-\frac{\delta_\lambda}{4!}\phi_r^4 $$ gives the following set of Feynman rules:
------------>------------ = $\frac{i}{p^2-m^2+i\epsilon}$
------------X------------ = $i(p^2\delta_Z-\delta_m)$
and the two 4-vertices.

The question is: Why look the Feynman rules for the first and the fourth term of the Lagrangian look so different? I believe the answer is connected to the fact that one has to bring the kinetic term of the Lagrangian to its canonical form $\frac{1}{2} (\partial_\mu\phi_r)^2$ and has to interpret everything else as (possibly momentum dependent) vertices. How does this look in formulae?