How does humidity affect the path of a bullet?
You are right. Field manual is wrong. Water has a lower molecular mass, which reduces air density.
But is the difference significant enough to notice?
Terminal velocity is the speed at which a falling body reaches a stable velocity when gravity and air resistance meet in a stable equilibrium. Let's assume the bullet travels close to terminal velocity so we can use its model.
Air density is part of the terminal velocity equation. Air density is inverse square-root proportional to terminal velocity. This means terminal velocity goes down, but only as a function of the square root of air density.
wiki - air density - humidity
Air density is charted above (credit to jaffer@MIT). You can see that in a hot climate of 30 deg C, air density has about a 3 percent from 0-100% humidity. But remember we have to take the square root of that difference - about 1.5%. So the most humidity can affect the speed of the bullet is 1.5%.
I don't know the math for how that affects a ballistic trajectory. Sorry.
It could be insignificant, which is why it they got it wrong. It's probably based on what people think intuitively, and backed up by confirmation bias. If it had a significant impact, you think they would know this by know. This scientific knowledge is not exactly new.
The field manual is wrong.
Today I have noticed that our humidity is extremely high compared to yesterday. The temp is about approx 5degrees F warmer today than yesterday. I just shot 30 rounds and noticed without any bias on this subject (no previous opinion) that my rounds were landing .125" higher than yesterdays 30 rounds. Hence why I am researching this subject of humidity and the impact on subsonic rounds. So for what it is worth, my experience has been humidity higher, POI higher. Then again, it could just be me.
According to several sources, such as http://longrangebpcr.com/accuracy.htm , you are right: higher humidity - higher point of impact. But those sources also point out that humidity effect is small.