Is this internal rearrangement of charges in a circuit a current and is this current different from the “normal” current of the circuit?
Is this internal rearrangement of the charges a current? 


Moving charge is a current, so yes, this is a current.

Also, the changing electric field caused by the rearranging charges is a current, called displacement current.  Whether a displacement current is "really" a current is a philosophical or semantic question rather than a physics or engineering question. From a physics and engineering point of view, displacement currents must be accounted for when solving the Kirchhoff's Current Law, they contribute to magnetic fields according to Ampere's law, and otherwise affect the rest of the world just as ordinary moving-charge currents do.


  If so then is this current different from the “normal” current of the circuit (The 1mA)?


It's a transient current (it falls to zero over time) where the "normal" current in this circuit will continue forever. 


  Electric fields that change and exist between charged bodies will cause something like current flow, called displacement current, such that a change in the electric field on one body affects the electric field on a nearby body. But how does this affect the 1mA current that appears in the circuit?


If your voltage source has internal resistance (like all real-world voltage source) then the extra current required to charge the parasitic capacitors will cause the source output voltage to drop, reducing the current through the 9 kohm resistor.

This effect will last for only a few nanoseconds or microseconds, until the capacitors are charged, and after that the presence of the parasitic capacitors will not affect the current through the resistor.

Note that in the model drawn, only C3 has its voltage changed when the switch is closed, so only C3 will draw current. The other capacitors won't cause any currents in response to the switch closing or opening.