The physics
A body in a liquid pushes some liquid aside. The liquid pushes back up on it. This push is the buoyancy. It equals the weight of the liquid pushed aside. That is Archimedes' principle.
B = ρliquid Vunder g
A body under water on a spring balance reads less. The balance holds only the weight minus the buoyancy.
A floating body sinks until the buoyancy equals its weight. So the part under the surface is ρbody / ρliquid. A loaded boat carries the most when the water reaches its rim.
Vunder / V = ρbody / ρliquid ρV = M + m
A light body let go under water feels a net push B − W. With no drag, that push throws it out of the water. A spring that holds it down feels the same net push.
(B − W) d = mgH kx = B − W
A Cartesian diver holds trapped air. Deeper down, the air shrinks, so the diver pushes aside less water. It hovers at one depth only.