Also your little animation is completely invalid. The light of the sun would cover the entire Earth, if the Earth was flat. In fact, even if the Earth was 100x larger than it is, sunlight would still cover its entire surface.
The question, "how is it that the earth is not at all times illuminated all over its surface, seeing that the sun is always several hundred miles above it?" may be answered as follows:--
First, if no atmosphere existed, no doubt the light of the sun would diffuse over the whole earth at once, and alternations of light and darkness could not exist.
Secondly, as the earth is covered with an atmosphere of many miles in depth, the density of which gradually increases downwards to the surface, all the rays of light except those which are vertical, as they enter the upper stratum of air are arrested in their course of diffusion, and by refraction bent downwards towards the earth; as this takes place in all directions round the sun--equally where density and other conditions are equal, and vice versā--the effect is a comparatively distinct disc of sun-light.