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...a side wind also changes vertical impact, with the amount of vertical change depending on the gyroscopic stability factor of the bullet.
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That's correct. A wind from the side will either raise or lower the point of impact based on the bullet's direction of spin. The lift or drop is caused by the Magnus Effect.
A clockwise spinning bullet (with respect to the shooter) will tend to impact lower on the target in a right to left wind and higher in a left to right wind. It reverses if the bullet is spinning the other way.
The simplest way to think about it is that the spinning bullet drags a very thin layer of air around it as it spins. If the bullet is spinning clockwise (with respect to the shooter) and the wind is from the right, that spinning air will tend to "bunch up" at the top of the bullet because of the wind and will tend to move faster at the bottom of the bullet because it's going the same direction as the wind. That raises the pressure above the bullet and lowers it below the bullet and that makes the bullet tend to fall a little faster than it would otherwise.
It's not a huge effect, but it can make a difference as the range stretches.