Let me explain it another way.
if you would fill a bottle with water and put it in water it would probably not move much ?
Why because the water inside is about the same weight than the water outside.
If you were to fill it with air, it would start to float to the surface ? Because the inside of the bottle is lighter than the same amount/volume of water ? This is why a balloon filled with helium would float at the surface: because helium is lighter than air.
In the same way if you were to fill it with lead it would fall in water because 1 liter of lead is heavier than 1 liter of water.
Now why can you go down when you exhale ? This is because you make yourself smaller by removing some air from your body. The weight stays nearly the same but the volume is smaller. Imagine that you would compress the bottle of the example earlier ...
Another way to explain it: a liter of water is 1 kilogram. If you fill the bottle with water, we saw that it would be about neutrally buoyant. Now if you empty the bottle and put 1kg of metal inside ,and close it, it would stay neutrally buoyant because the bottle is still 1 liter of volume and still weigh 1kg.
Now you could suck out the air out of the plastic bottle and leave 1kg of metal inside with no air at all. It would start to drop because even though it weighs one kg it is now smaller than 1kg of water: obviously metal is heavier than water ... When you exhale and makes yourself ‘smaller’ this is what happens.
Now back to when you breathe air out of your tank. It’s like taking some of the metal inside the bottle out. Your tank is the same volume but weighs less: air is light but still have some weight and when you compress is many times it is not negligible.
