I believe that’s the same for every planet. And every moon. For every orbit.

Its just that the barycenter is inside the more massive object when one is much more massive than the other. Not that this makes much of a difference to anything.

I mean, sure, but that’d be like saying I’m pulling the earth towards me when I jump.
If you have ever done a handstand then you have lifted over your head the weight that the entire mass of the earth has in your own gravitational field.