For the sealed bucket with "vacuum" or "low pressure air", are we talking about it sealed to itself or sealed to the bottom of the pool? I think it's two different scenarios. For it sealed to itself, I think we wouldn't notice much difference in the upward force whether the bucket was sealed with "sea level air", "vacuum", or "low pressure air". The variation of upward force could be compared between the three as minor (as a percentage), but the total force in all three cases is strong. I will avoid explaining why this is the case with accepted models, and stick to verifiable experimental evidence.
I never mentioned a lid. I would mention a lid if I need to go into that.
For the case where an inverted, lidless bucket is sealed to the bottom of the pool with vacuum, or "low pressure air", I don't think a swimmer would have a chance in hell of removing it from the bottom with brute force. In that case there is a lot of pressure on the outside of the bucket pushing it downwards, and essentially none on the inside pushing it upwards. The catch here is that this would only work in practice if the air/water were pumped out of the bucket after it was submerged and sealed.
Of course, which is why I likened it to an evacuation chamber in atmosphere.
I'll add a third case for consideration. If the bucket is inverted and submerged with "sea level" air pressure and then pushed to the bottom of the pool, something interesting happens. The air in the bucket compresses, and without air bubbles leaving the bucket, some water will make its way into the bucket. You can invert your thinking, and view this as the "air level" of the bucket decreasing from full to partially full. The interesting thing is that no air has left the container, so the same amount of air is now taking up a smaller space. Maybe i went too far, but let's see what you have to say about it.
Yep but the issue is in using energy to actually push that inverted bucket down.
You see only energy applied will compress the air against the water.
The thing is we need to get to where we know that atmosphere pushes back against the dense objects displacement of it and we are at a point of using water as the analogy to show why it happens.
People can't seem to understand the atmospheric push/squeeze down against any dense object that also displaces it's own dense mass of atmosphere that overcomes the resistance of atmosphere under it, which is why it sits on a foundation in the first place.
It overcomes buoyancy.
This is about seeing how denpressure would create a scale weight measurement instead of so called gravity creating it.
Placing a scale plate on the bottom of the water pool then sitting the bucket onto it full of water, we get the measurement of the bucket skin. the dense mass of the bucket itself, not the mass of water because the water in the bucket is equalised with the surrounding water, just as if it was on the ground against atmospheric pressure, except for different dense masses between water and air, obviously.
So therefore you get a scale reading of the bucket by the displacement of water by the skin of the bucket and it's thickness.
This displacement is pushed back into the pool, obviously and it's this amount that is directly measured back to the scale plate.
To make it more visual in mind. Imagine the bucket is simply squashed into a dense small block. Let's say 1 cubic centimetre for the sake of it.
This would be the amount of displaced water that would be added to upset the equilibrium to create the weight measurement on the scale plate.