Does Lorentz's force and magnetic dipole alignment torque explain same or different forces inside an electric motor?

I had hoped to see answers to your question, but since this is not the case, I allow my own answer here.

The torque in your left image can be explained as follows:

• Electrons have a magnetic dipole moment. In a wire without an external magnetic field, the magnetic dipoles of the electrons are randomly oriented.
• When the electrons are in a magnetic field, their magnetic fields get aligned with the external field, and the loop behaves like the magnet in your right picture. It rotates (clockwise) and comes to rest with the wires close to the poles. The magnetization of the wire depends from the susceptibility of the wires material.
• When a small current is switched on, the Lorentz force acts against the magnetic alignment and the wires move slightly away from the vertical position. What you observe is that the movement of the electrons within an external magnetic field is responsible for the lateral movement of the wire. The Lorentz force acts against magnetic alignment.
• When more electrical power is used, the rotor starts to rotate.

So

Do these two explanations describe two separate forces that are accumulated when calculating the torque of a motor ...

is the right answer.