DC Motor Speed and Current
Can someone explain with a little more intuition on how applied voltage determines the motor speed rather than the current through the motor coils?

Any permanent-magnet motor (DC, brushless DC, or brushless AC) is also a generator.  It generates a voltage that's proportional to the motor speed times a constant that depends on the magnetic field intensity, the number of turns of wire in the coils, and the geometry of the motor.
When a motor is turning, this generator effect is also active, and the generated voltage is in opposition to the applied voltage.  It's called back EMF*.  If you had a perfect motor with either no friction or no resistance in the coil windings, the motor would spin up until the back EMF exactly equaled the applied voltage.

I read that the rated voltage is the parameter that defines the speed of the motor.

Well, no.  If you know the motor's voltage/speed constant, and it's a typical motor, then when it's running unleaded its speed is pretty close to the applied voltage by the voltage/speed constant.  Basically, unloaded it acts a lot like an ideal motor.

Is there any way to calculate the current through the motor coils?

Yes.  You need to start with the motor speed, the motor's speed/voltage constant, and the applied voltage.
So, calculate the back EMF: \$v_a = \omega_a k_t\$, where \$\omega_a\$ is the armature speed in radians per second, and \$k_t\$ is the voltage/speed** constant.
The effective voltage on the coils is the terminal voltage less the back EMF.  So you use that and Ohm's law to find the current: $$i_a = \frac{v_t - v_a}{R_a}$$

Is the above answer same for DC motor and stepper motor as well?

Not really.  A stepper motor does generate back-EMF, and there is a lot of overlap between it and a DC motor.  But basically a stepper motor goes at the step rate, until the torque it needs is greater than the torque it can supply -- then it slips, and you say bad words.
* &quot;EMF&quot; = &quot;electro-motive force&quot;, which is the &quot;correct&quot; term for &quot;voltage&quot;.
** I'm using the symbol for the torque / current constant, because in an ideal motor they're the same, and I'm lazy.