So in an electric motor the torque is generated from the Lorentz force on the current carrying wire by the interaction with the outer magnetic field. There is also another interaction between the magnetic dipole created by the solenoid (or wire) and the external magnetic field which would drive the motor as the B-fields repel and attract each other periodically.
The torque which you get via the Lorentz force comes out to be $BIA$ where $A$ is the area of the coil ie length $L$ times the width of the coil.
The magnetic dipole moment of the coil is $IA$ and the torque is the cross product of the dipole moment and the magnetic field which works out to be $IA\,B$ with the orientation of the coil and magnetic field as shown in your diagram, which is exactly the same magnitude for the torque as found using the Lorentz force.
It is two different ways of looking at the situation and getting the same result not two different ways of a torque being applied to the coil.
There is also another interaction between the magnetic dipole created by the solenoid (or wire) and the external magnetic field which would drive the motor as the B-fields repel and attract each other periodically.
Your statement, that the rotor is a coil of a current carrying wire and a magnetic field is - beside the external magnetic field -induced, is fully reasonable.
Is an electric motor driven by both these forces or are they the same force?
The external magnetic field reinforces this magnetic field on one side of the coil and weakens it on the other.
Overall, this leads to the fact that the neutral are of the commutator, in which the polarity of the current must be switched over, has to be a little bit shifted in the direction of rotation. Otherwise the motor operates every 180° for a little time against the direction of rotation in generator mode.
Furthermore the induction of this shifted field led to the induction of currents and this to sparks in the commutator.