  The 1538 nm ${}^4$I$_{15/2}$ - ${}^4$I$_{13/2}$ transition of Er$^{3+}$ has
an unusual hybrid electric-magnetic dipole character, and signatures of the
hybrid moment can be expected in coherent transient measurements. Here, we
investigate the effect of the hybrid moment in both sites of
Er$^{3+}$:Y$_2$SiO$_5$ on Stark-modulated photon echo measurements, showing
that it results in a reduction of visibility of the modulated signal as well as
phase and polarization shifts. We interpret these effects using a simple
optical Bloch equation model, showing that site 1 has a strongly mixed moment
and site 2 is predominantly magnetic dipole. We discuss the implications of the
hybrid moment for quantum information applications of quantum memories. We also
use the echo measurements to extract the Stark shift of the optical transition
along three orthogonal directions, finding values between 10.50 and 11.93
kHz/(V/cm) for site 1 and 1.61 and 15.35 kHz/(V/cm) for site 2. We observe a
modification of the Zeeman shift by the electric field in both sites and
discuss how this may be used to electrically control Er$^{3+}$ spin qubits.
