Do not confuse displacement and buoyancy. They are related, but not the same thing. The reason that the empty jar floats is because the volume of air in the container weighs less than the volume of water that it would displace. This is what keeps steel ships from sinking.
I'm not confusing displacement with buoyancy, it's you lot that are doing exactly that whether by naivety or by deliberate means.
Basically you're being duped.
You're answering your own questions against yourselves and can't see it because your minds are focused on twisted experiments.
Why would you want to compress the air inside? If you're in a submarine, then you may want to increase your density, but you don't really want to compress the air (or the human occupants) inside.
It's not about crushing the occupants inside. It's about allowing the sub to sink to a depth that it can no longer push further than.
If it does this it will be squeezed and so will trapped air inside.
The more it is squeezed the deeper it goes until it no longer holds atmosphere.
From this point on it gives a proper density displacement.
The only other way to displace water by the same equal manner is to have the sub above water, whether that is high above or extremely low.
This is displacement by buoyancy, which creates the exact same displacement as what I said in the depth sinking.
Is that mixing up buoyancy and displacement.
Not really.
Volume is about the buoyancy and the density of That volume is what displaces the water or the atmosphere.