I think the post's confusion comes from trying to focus on where the shadow falls, while the concept of eclipse really is focused on what is obscured from view. I.e. calling them lunar and solar is consistent in the later sense.

In case of lunar eclipse, the view of the Moon is obscured by the shadow of Earth. I.e. Sun's light is blocked by Earth, so it doesn't reflect back to Earth from the Moon. In case of solar eclipse, the view of the Sun is obscured by the Moon itself (which throws shadow on Earth), i.e. you don't see direct light from the Sun blocked by the Moon.

Both are pretty real and both are called eclipses.