How does a load decide the amount of current drawn in a circuit?
why do we say "current drawn depends on the type of load connected


We actually say that the current drawn depends on the load AND the power supply voltage. In other words, what we should say matches ohms law.


  Moreover does the above reasoning of using ohm's law hold valid for 
  any type of load? i.e. L load, C load, RL load or a combination of all 3)


No, it becomes more complex with capacitors and inductors because the current drawn by a capacitor depends on the rate of change of applied voltage and not the absolute value of voltage.

For an inductor, the current depends on the integral of voltage with respect to time hence, inductor current is proportional to applied voltage x time.

That’s for DC circuits. It’s another departure from ohms law for AC circuits and handling complex transients but the same rate of change (or integral) principles apply.