The equations of motion for how bodies move in the Solar System, which are then fitted to the observational data of positions and ranges to provide the ephemeris, involve a nested set of effects which account for increasingly subtle and smaller effects.
As detailed in the documentation for DE430 and DE431 and the introduction in Section III these are:
- the basic N-body gravitational attraction between all the bodies, treated as point-masses
- the effects of the non-spherical oblateness of the Sun (its figure as it is described) on the other bodies of the Solar System
- the effects of the static non-spherical shape of the Earth and the Moon on each other and on the planets Mercury to Jupiter
- the effects of the time-varying shape (tides) raised on the Earth by the Sun and the Moon back on the Moon's orbit.
For 1. this is a generalized version of the classical force/acceleration due to 2 bodies $F=\frac{Gm_1m_2}{r^2}$ (e.g as in these course notes) but extended to include multiple (N) bodies (Newtonian N-body equations of motion) and generalized beyond the effects of Newtonian gravity to allow general relativity to be included (the so-called parametrized post-Newtonian (PPN) metric). This acceleration on a particular body is summed over everything else: the Sun, the Moon, the planets Mercury through Pluto and the 343 largest asteroids. So this is where the statement you quote
However, at the JPL Horizons website it is said, that the effects of 8 planets are considered.
comes from as all the planets (plus more) are included in the basic equations of force/acceleration.
In addition to the basic equations from 1., the effects of non-spherical, non-point mass bodies are included as detailed in Section III B and which you quote in your question. These effects are:
- the non-spherical Earth (up to 4th degree in the spherical harmonics expansion of the non-spherical Earth) on the Moon, the Sun, the planets Mercury - Jupiter (all treated as point masses)
- the non-spherical Moon (up to 6th degree) on the Earth, the Sun, the planets Mercury - Jupiter (all treated as point masses)
- the effect of the second order oblateness of the Sun on everything else
These effects are going to be much smaller than the main gravitational effect from 1. For example we very rarely need to take into account the $J_2$ effect of the Earth when calculating the effects on Near Earth Object trajectories and this is the largest of the non-spherical effects (the higher harmonics are weaker still). An additional issue is that we don't have very good gravity data that would reveal higher harmonics for the outer planets as this can normally only be measured by close-orbiting spacecraft and Uranus, Neptune and Pluto have only received brief distant flybys. (I suspect additional gravity data may be coming out for Saturn based on the 'Grand Finale' orbits of the Cassini spacecraft but this is likely still being worked based on these abstracts)