Only if either the Earth is in synchronous rotation around the Sun or the Moon is in synchronous rotation around the Earth will the caption from the third figure “Lunar eclipse observations cannot determine the Earth’s sphericity on their own.” be true.
In the first case, people who could observe a lunar eclipse would be in permanent night and have to infer the existence of a source of light, and they would be aware of the shape of the Moon from observations of its day/night boundary. In the second case, only half the population would even know of the moon’s existence through direct visual observation and they could consistently assume the Moon too is flat.
But this is probably physically impossible. I think most of the following is reasonably correct: In the first case, if a high spin is required to maintain oblateness of the Earth, and the axis is always pointing to the Sun, then some large torque is changing the largest angular momentum component. And that torque has to be in the ecliptic (as a covector) or “perpendicular” to it if you insist on thinking of torque as a vector. So the Sun would have to be lobed? In the second case, if the Earth is flat and the Moon is synchronous (meaning that large angular momentum component of the Earth is pointing towards the Moon) the Moon would have to be large and lobed to produce the required torque and Earth would either be the satellite or they would be “twins” like Pluto and Charon.
Physics places quite rigid constraints on speculation on what the world could be like, even plain old simple classical mechanics.