A pellet gun is not an air gun.
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Perhaps technically speaking this is correct. An air gun much like an Air-Soft gun usually shoots a plastic ball. While it is potentially dangerous the speed and weight of the projectile is slow and light enough that it would only cause damage to a person hit squarely in the eye.
A Pellet gun also uses air to project the projectile but the projectile is usually a pellet or round ball made of lead in a wide range of sizes from .177 to over 50 Cal. The projectile speed at discharge of a pellet gun can be more than the speed of many gunpowder guns. This speed and weight combination can be harmful or even fatal to humans. These guns, the heavier caliper versions anyway are often used as varmint control weapons.
All this being said, in the current status of the English language many people would easily believe that an air gun and pellet gun are the same thing.
Lewis and Clark carried repeating air rifles that were used to impress the Native Americans.