Motors in general have always been a difficult subject that I cannot fully wrap my head around. Considering DC motors, what determines the rate at which the motor spins?

It was my understanding that a permanent magnet created the field by which the current through the motor would act against via the conductor's induced field. As current increases then the induced field would thus increase - thereby increasing the rotational speed.

https://i.stack.imgur.com/s37EI.jpg

However, I have read quite a bit of material that has led me to realized that I was incorrect. Namely, what is said at this link about DC motors: http://farside.ph.utexas.edu/teaching/302l/lectures/node93.html. And I've got many components from kynix: http://www.kynix.com/ to be more practical.

For example, the same circuit schematic as above produces (considering back emf) the governing equations:

https://i.stack.imgur.com/dYNhd.jpg

So we have the current through the motor as a function of the back emf.

Is the back emf a function of the load on the motor? Is it that the emf is generated in such a manner that current is limited by the lessening of the potential difference between it and the supplied voltage?

The governing equations dictate that if the applied voltage is lowered then the back emf decreases further which in turn will decrease the current demanded by the circuit (through the motor).

So, is the current through the motor just an indirect indication of the speed, or how does the current otherwise affect the operation?

Are all DC motors (aside from brushless) similar?