The point that gravity from the moon has a similar effect as gravity from the earth is about 80% of the way to the moon. This means that if something is 79% of the way to the moon and stationary, it will fall towards the earth.
Atmosphere is pretty much 1% to the moon or less (if you don’t count the exosphere, it’s only .2%). If any is escaping earth’s gravity, it’s not because of the moon.
An object with greater mass has a greater gravitational pull, and the effects of gravity decrease dramatically with distance. The Earth is much larger than the moon, so it pulls much harder. And, the atmosphere is much closer to Earth than it is to the moon. So, the moon DOES pull on and deform the atmosphere (just as it does with the water in the oceans to create the tides), but the attraction to the Earth is far too strong for the moon to have any chance at pulling the atmosphere away.
The moon does cause atmospheric tides in the atmosphere, you just don’t notice it much for the same reason that creatures on the ocean floor generally don’t notice the tides.
The tides are not air and water being pulled towards the moon, but more being pulled down to the center of earth less in some places than others.
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