"Precession of the equator" refers to long-term changes in the Earth's rotation axis relative to the mean (average) orbital plane of the Earth-Moon barycenter about the Sun, aka the ecliptic. "Precession of the ecliptic" refers to long-term changes in the mean orbital plane itself.
The Earth's orientation with respect to the ecliptic changes primarily because of torques exerted on the Earth by the Moon and the Sun, but also because of torques exerted by other planets. Because of those other torques, it is not quite correct to call this "lunisolar precession".
The mean orbital plane of the Earth-Moon barycenter about the Sun also changes slowly due to the influences of other planets (and also the Sun's not-quite spherical shape, and also relativity). The mean orbital plane of the Earth-Moon barycenter about the Sun is tilted a bit (about 1.57°) from the solar system invariable plane.
There are also short term effects, with frequencies less than 20 years. These short term effects are lumped into two groups. Nutation models the short term effects that can be predicted for hundreds of years. Like general precession, nutation also results from torques on the Earth's rotation and its orbit. Polar motion describes short term effects that cannot be modeled accurately beyond a year or so. Polar motion includes what physicists call torque-free precession. Polar motion also includes effects due to transfer of angular momentum between the Earth's core, mantle, oceans, and atmosphere. These small amplitude motions are observed. Predictive models don't cut it after a year.
From the perspective of physics, precession and nutation are aspects the same phenomenon. The frequency gap between the Earth's precession and nutation is so vast that it makes sense to model them separately.
In theory, other stars also exert an influence on both the Earth's orientation and its orbital plane, but these effects are so small that they are not modeled.