However, from viewing the motions of the planets above us, it does not necessarily follow that the Earth is a planet in the Solar System. The Earth is a fundamentally different kind of body; a plane rather than a planet
I'm curious to know why the FE movement would think this way? To me the way the planets move through the sky does not suggest in any way at all that the Earth is a plane. All the planets follow a path through the sky along the ecliptic which is shared closely by the Moon and the Sun. Why would that make it seem like the Earth is a plane?
Rather what we see suggests that the planets are all orbiting the Sun in the same or very similarly inclined planes and the Moons orbit around the Earth inclined at just 5 degrees from the plane of the Earths orbit (i.e. the ecliptic).