What does the phrase "limb of the earth" or "atmospheric limb" mean?
The etymology of limb for an astral object comes from the original Latin root. Limbus means border in Latin, a language in which all good students of last centuries were proficient, including astronomers; when describing heavenly bodies viewed through the telescopes they shortened it to limb.
- (astronomy) The apparent visual edge of a celestial body.
- (on a measuring instrument) The graduated edge of a circle or arc
Now limb as an appendage of a body or tree has a different etymology in English. It comes from Middle English lim and acquired the "b" later.
The limb of the Earth is simply the horizon, or the edge of the Earth at the horizon, especially if you see it from a spacecraft. When you also explicitly see the thin layer near the horizon corresponding to the atmosphere, it's the atmospheric limb.
There's no problem with the Earth's being spherical because the spherical symmetry is respected by both phrases containing "limb". On the other hand, the "limbs" are limb-like because in the one-dimensional model of the Earth where the altitude is the only coordinate, the limb (in both senses) is at the end point, much like hands and feet are "end points of a body".