  A deeper insight into electron elastic scattering off endohedral fullerenes
$A$@C$_{60}$ is provided. The study accounts for polarization of both the
encapsulated atom $A$ and C$_{60}$ cage by an incident electron. It is carried
out in the framework of the combination of both a model and the first-principle
approximations. A core principle of the model is that the C$_{60}$ cage itself
is presented by an attractive spherical potential of a certain inner radius,
thickness, and depth. The main idea of the first-principle approximation is
that the polarization of $A$@C$_{60}$ by an incident electron is accounted with
the help of the Dyson equation for the self-energy part of the Green's function
of an incident electron. Calculations are performed for, and comparison is made
between the individual cases: (a) when $A$@C$_{60}$ is regarded as a static
system, (b) when C$_{60}$ is a static cage, but the encapsulated atom is
polarizable, (c) when both C$_{60}$ and $A$ are polarized simultaneously but
independently of each other, and (d), as the most general scenario, when
polarizabilities of C$_{60}$ and $A$ are coupled. Spectacular similarities and
discrepancies between results of each of the exploited approximations are
demonstrated. Revealed features of the overall significant impact of
$A$@C$_{60}$ polarization by an incident electron on its elastic scattering off
$A$@C$_{60}$ are demonstrated. Choosing Ne, Xe, and Ba as "probing" atoms, the
dependence of $e + A@{\rm C_{60}}$ scattering on the size and polarizability of
the encapsulated atom is unraveled.
