I want to know, by definition, does the wind include the air being moved, and thus have a weight, or is it only the kinetic energy, and thus doesn't have a weight.

I hope that this, very simplified answer will make you more interested in meteorology and Earth science in general.

Whether dictionary definitions are important in the context and spirit of a riddle is, of course, up for debate :-)

So the mass of air plus the mass of wind is the same as the mass of air. Therefore, the mass of wind must be 0:

m_air + m_wind = m_air

m_wind = 0

_{*) Assuming the volume is large enough that the effects on the border of the volume are relatively small enough.}