[RonF]

The point about which both the Moon and the Earth orbit about each other stays about 1000 miles beneath the Earth's surface (but obviously it's location changes as the two bodies orbit and the Earth spins on it's axis). In such a case, I'd hold that the Moon orbits the Earth. If the point about which two bodies orbit each other is not beneath the surface of one of them, but is between them, then I'd call that a binary system.

[beezdotcom]

The point about which both the Moon and the Earth orbit about each other stays about 1000 miles beneath the Earth's surface

Oh YEAH? Say that to my FACE (in about ~10⁸ years or so, when I might actually be RIGHT...gotta remember to watch those decimals...)

[Doctor Stochastic]

If the point about which two bodies orbit each other is not beneath the surface of one of them, but is between them, then I'd call that a binary system.

Sometimes the Sun-Jupiter system has its center-of-gravity outside the Sun's surface.

[Gorjus]

If the point about which two bodies orbit each other is not beneath the surface of one of them,...then I'd call that...

Well, as you said, it's what you would call it, and you're free to do whatever you want. It's a logical distinction, and there are worse.

However, I think a better one is to say that if the gravitational attraction of some other body (like a nearby planet) is greater than the gravitational attraction of the star, then it's a moon. Otherwise, it's a planet (assuming it's also big enough to be round).

By that definition, our Luna is still a planet, since if you work the math the gravitational attraction of the Sun on Luna is greater than the gravitational attraction of the Earth on Luna.