You're right Erasmus, but..
Consider two planets of an hypothetical solar system, two planets in the same orbit and pretty close to the other, revolving one around the other one.
Question: Wich one will revolve?
Answer: the two. They will be revolving at the same time around the other one.
Do you agree with that?
OK
But do the two planets have exactly the same mass?
I guess not (there always be a difference, at least small, but they won't be equal in mass.
But then, should the smaller planet revolve around the greatter one?
Yes, and no. We still could see the two planets revolving around the other one.
But now consider that the two planets have different masses, one bigger and one smaller.
What would we see? We'll see the smaller one revolving, and the greater one tottering.
But it is the same effect amplified. We can see it as the two planets revolving, or one revolving and the greater one just tottering.
The same case is for the earth-moon system, but in this case the moon has far less mass than the earth.