Of course, and that’s why an observer in another planet would say: “look, an Earth eclipse” (or at least they would if the Moon was big enough for the event to be easily noticeable for them or if they paid enough attention, but that’s not the main point).

What we call “a lunar eclipse” is a solar eclipse in the Moon. What we call a solar eclipse is a solar eclipse on Earth. We can also observe solar eclipses (total or partial) on other bodies in the solar system.

I agree that both are real, with a real shadow. But for a (total) lunar eclipse the full Moon is in the shadows, while in a solar eclipse a little shadow traverses the Earth surface.

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.

For what it’s worth, we also call eclipses those that happen in Jupiter when one of its large moons is between Jupiter and the Sun. Those are also real, and allowed for the first estimate of the speed of light: https://en.m.wikipedia.org/wiki/Rømer%27s_determination_of_t...