I have been doing some research on the Lorentz force law because I am currently learning about AC and DC electric motors. I have been stumbling onto the phrase "electromagnetic force" and "Lorentz force." Currently, I think that the Lorentz force law is what explains the generation of motional EMF in an electric motor, but I am not 100% sure.
Are they the same thing or are they somehow related?
The words are often mis-used. It's probably better to express exactly what's meant with equations, than to try to insist that everybody uses the words according to their dictionary definitions.
Lorentz Force is that force on a moving charged particle, due to the electric and magnetic fields in the region it's moving through.
$$ \boldsymbol{F = q(E + v\times B)}$$
However, a lot of people say Lorentz force, when they really mean the force due to just flowing current or motion of charged particles in the magnetic field, using only the \$ \boldsymbol{qv\times B}\$ term, which is more strictly called the Laplace Force, when it's had a trivial manipulation applied to it to turn qv and a length of conductor into a current.
If we talk about Electromagnetic Force, then we can mean any or all forces that arise from the interaction of charged particles, electric and magnetic fields (near fields), electromagnetic fields (far fields), electric charges and flowing currents, in any combination, according to our context.
The direction of the induced EMF in a motor is most simply given by Lenz's Law, which qualitatively says that the direction is such to oppose the motion that's causing the induction.
The magnitude and direction of the induced EMF in a motor is given by Faraday's Law of Induction.
These latter two 'laws' are included in, or are derivable from, Maxwell's equations. The have separate names due to how they were discovered, historically. We still keep these names today, mainly for their didactic simplicity. For instance, a grade school pupil can understand Lenz's Law quite easily, whereas they need to be more mature to handle the quantitative laws. I remember being quite young when introduced to 'the Law of Sheer-Cussedness', aka Lenz's Law.
The electromagnetic "force" is the conventional name for one of the four "forces" of nature. I.e. it refers to a huge load of (all in fact) effects that arise due to the electronic charge of matter. The Maxwell Equations are usually considered to be the fundamental description of classical Electromagnetism.
The Lorentz force is a specific emergent effect felt by a massive particle moving through a magnetic field. It serves as a basis for the deduction of a range of phenomena, such as: electromagnetic induction or the (normal) Hall effect.
The creation of the EMF in a running motor/generator is usually explained using the laws of induction so it does depend at a deeper level on the Lorentz Force, as well as on Electromagnetism at a yet deeper level.
