Your points 1.-3. are exactly correct. It is also important to recognize that these two forces are external to the floating body. As such they could in principle exert a net force on the floating body, which would change its linear momentum in time, and they could exert a torque (moment of force), which would change its angular momentum in time.
The second important ingredient is, that the floating body is rigid, i.e., it does not deform, an it has an inertial moment.
Concerning the net force, it will be zero in both cases, the left and the right figure. The buoyancy force acts vertically up, and the gravitational force acts vertically down. The floating body will sink into the water until both forces are of the same magnitude. Since they act in opposite directions, the net force is zero, and the linear momentum of the floating body is constant in time, i.e., it is zero.
What differs between the two figures is the magnitude of the torque. In the left case, the two external forces act along the same vertical axis, and the resulting torque is therefore zero. In case of the right figure, the buoyancy force, which acts vertically upwards from point B is not collinear with the gravitational force, which acts vertically downwards from point G. The result is a torque that acts as to restore the original equilibrium position of the body shown on the left.
In order to bring the floating body into the tilted position shown on the right, some external agent had to do work against the torque that gradually built up. This means that the floating body has acquired some potential energy as compared to the equilibrium situation on the left. The external agent can achieve the tilt by applying an external torque without applying a net linear force.
Once this external torque is removed, the system starts to oscillate.
The remaining physics is that of a pendulum. Starting from the tilted position, the floating body will be set into rotating motion towards the equilibrium position. Doing this, potential energy is gradually converted into kinetic energy. When the body reaches the equilibrium position, the torque will be zero, but the body has all the potential energy converted into kinetic energy of rotation. It will therefore swing through the equilibrium position, again converting kinetic energy into potential energy, until the former is zero. For sufficiently small oscillation amplitudes, this oscillation will be harmonic.
In a realistic system, this motion will suffer from friction, which gradually converts kinetic energy into heat flowing off into the surrounding water. This situation corresponds to a damped harmonic oscillator.
This principle is responsible for the stability of ships against tilt. A ship will be the more stable, the lower point G is. This is why modern yachts have a lot of heavy mass in their keel. The restoring torque will also depend on the shape of the floating body, which will play a prominent role, if the tilt becomes larger and larger.
A more elaborate treatment can be found here.