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Actually, if you look at the orbit, "our" moon actually orbits the sun, not Earth. The moon's orbit is always concave toward the sun.

It depends on what is meant by "directly orbiting". This geometric definition is not valid. The wave formed by the motion of the moon around the earth being carried around the sun is an accident of the location of the earth. If you moved the earth far enough from the sun so that the moon's orbital velocity around the earth were faster than the earth's orbital velocity around the sun, then the moon would no longer fit that definition. The definition of a planet should not depend on its distance from the sun.

If the sun were gone, earth would still be a planet but the moon would not because it would remain in orbit around the earth (and earth is not a star). The moon could only be a planet then if the earth were also gone. Therefore, the moon is in direct orbit around the earth, not the sun.

Almost every moon is bound in orbit to the sun in the sense that if the planet suddenly disappeared, that moon would remain in the solar system. The only exceptions might be moons very close to Jupiter whose velocity around Jupiter combined with Jupiter's orbital velocity would exceed escape velocity from the sun at certain points in its orbit.

But I submit that any smaller object which would remain gravitationally bound to a primary object other than the sun if the sun were gone has an intermediary and is not in "direct orbit" around the sun and is not eligible to be designated a planet. (Certain objects that are merely in orbital resonance with another body around the sun, which would escape that primary body if not for the sun's gravity, such as certain Apollo asteroids which come around one side of the earth and then the other forming an extremely elongated and unstable temporary "orbit" but would fly away if not for the sun bringing it back are in "direct orbit" around the sun and not the earth.)

(Certain objects that are merely in orbital resonance with another body around the sun, which would escape that primary body if not for the sun's gravity, such as certain Apollo asteroids which come around one side of the earth and then the other forming an extremely elongated and unstable temporary "orbit" but would fly away if not for the sun bringing it back are in "direct orbit" around the sun and not the earth.)

Example: Cruithne

This geometric definition is not valid.

I guess we'll just disagree. While you can make the case that being or not being a 'wanderer' is independent of distance from the sun (so that 'loose' planets far from any star are still planets, and not rocks or whatever), the issue is whether Luna is a planet or a moon. If you took the earth away, then Luna would be a planet at any distance from the sun, so is your definition that distance from the sun doesn't make a body a planet, but distance from another body does?

My definition of the difference between moon and planet would be based on whether the gravitational attraction of the nearby 'planet' (in this case Earth) on the smaller body is greater or less than the gravitational attraction of the sun on that smaller body. In the case of Luna, the sun exerts a greater force than the Earth does.

Obviously, this sort of distinction - whether a particular body is considered a moon or a planet - depends exactly on whether there is another nearby planet which exerts a greater force on the body than the sun itself.

By that definition, though the moons of Jupiter and Saturn would continue to orbit the sun if the planet were removed, they are nonetheless moons because the force on them from their planet is greater than the force on them from the sun.

And this can be recognized by whether their motion is ever retrograde with respect to the sun or whether their motion is ever convex toward the sun.

You're welcome to your own definition any way you want to make it, but I think the gravitational force definition makes more sense. You are, of course, free to disagree.