I ran into this problem when I want to simulate subsatellite points.

There are generally two kinds of descriptions or definitions of the subsatellite point:

1. The intersection point with the Earth surface by the line from the Earth center to the satellite.
   • This definition is problematic because
     • it doesn't define what is the “Earth surface”, and it seems assuming a spherical Earth here;
     • it actually assumes geocentric latitude by this definition.
   • But many places use this definition, such as
     • (NASA Earth Observatory Glossary) subsatellite point: Point where a straight line drawn from a satellite to the center of the Earth intersects the Earth's surface.
2. The point directly below the satellite. This is still confusing because it doesn't define what is “below”.
   • A more precise expression may be (see here, CelesTrak), “it is that point on the Earth's surface where the satellite would appear at the zenith.”

Wikipedia even gives a contradict description (https://en.wikipedia.org/wiki/Ground_track):

A satellite ground track may be thought of as a path along the Earth's surface which traces the movement of an imaginary line between the satellite and the center of the Earth. (This obviously use the first definition.) In other words, the ground track is the set of points at which the satellite will pass directly overhead, or cross the zenith, in the frame of reference of a ground observer. (but this obviously relies on “zenith”, the second definition above.)

https://space.stackexchange.com/a/19729/28220

For Earth observation satellites, “nadir” points toward the geodetic subsatellite point. The geodetic subsatellite point is that point on the surface of the reference ellipsoid that is closest to the satellite. This is equivalent to the point on the ellipsoid where the satellite is at zenith (“straight up”). Using nadir as pointing to the center of the Earth will miss geodetic subsatellite point by up to 21 km for a satellite in a sun synchronous orbit. (I should use this one for ground track simulation.)

For satellites formation flying, and rendezvous and proximity operations, “nadir” and zenith point to / away from the center of the Earth, in the LVLH frame. This is primarily because of the use of the linearized Clohessy-Wiltshire equations in these fields.

An even more subtle meaning is that nadir points in the direction of the point on the geoid that is closest to the satellite. (see original post. I don't understand)