If we approximate "air" and "water vapor" by treating them as ideal gasses (which is an assumption that is met within a truly miniscule fraction of a percent under normal atmospheric temperatures and pressures) the the density (weight per unit volume) of each gas in the mixture is proportional to its Atomic Weight.
Water vapor is easy, that portion has an Atomic Weight of 18 (one Oxygen atom at 16 and two Hydrogen atoms at 1 each).
Air is a more elaborate mixture but for this discussion can be treated as a mixture 79% Nitrogen (Atomic Weight 28, two atoms at 14 each) and 21% Oxygen (Atomic Weight 32, two atoms at 16 each). So each of the major components of dry air weighs more per unit volume than does the water vapor.
Basically, in a normal mixture of gasses the proportion of molecules of each species and the proportion of the density represented by each species is the same. That's because the behavior of gasses (unless the pressure is very, very, very high) is to spread out where each unit of volume contains the same number of molecules, regardless of the species (element) of each molecule.
So humid air is a mixture of dry air molecules with much lighter molecules of water vapor, resulting in lower density for the humid air.
P.S. I believe the anonymous R&A member is being misquoted. The flight of a golf ball, aerodynamically, is definitely different depending on air temperature. Colder air is more dense therefore the ball will not fly as far. The recent findings being (mis)quoted are concerning the effect of temperature on the springiness and resulting ball speed of a solid-core golf ball at various temperatures. Older balls with wound cores were known to compress and rebound somewhat sluggishly at low temperatures so it really made sense on a cold day to do the Tom Watson trick of keeping balls in a pocket next to a hand warmer. What they've found now with modern balls is that the compression and rebound performance only varies trivially over a normal temperature range of something like 0-40C.
So the warm-day and cold-day performance of the ball used to be different due to two effects. The effect of cold or warm air on the flight of the ball and the effect of warm or cold windings and cover in the ball itself. The latter effect is now negligible so only the aerodynamic effect remains to any meaningful extent. But that effect is same as it ever was.