Aren't "alt-text" and "image description" the same?

No, they aren't.

That is, alt-text must include an image description. But an image description is not always an alt-text.

It's only an alt-text if it takes the place of the image if the image cannot be seen, and literally so. Alt-text can be best understood as an image description hidden underneath the image. For those who can't see the image for any reason, the alt-text emerges.

However, image descriptions can exist elsewhere than in the alt-text. For example, an image can additionally be described in the post itself. But a description in the post text isn't an alt-text because it's next to the image and not "underneath" it. And even then, there should always be an image description in the alt-text.