I know, that in a reversed biased connection the reverse saturation current flowing is negligible but before the depletion layer is widened to its maximum capacity. It seems the current flowing through the circuit would have a great magnitude (depending on the power source). Is there any name given to that current?
As stated in the comments, the depletion layer acts like the insulating dielectric layer in a capacitor. The effect is much like charging a capacitor through a resistor (the rest of the diode), except that as the layer widens the capacitance falls. The current has no special name.
This variation in capacitance with applied reverse DC bias voltage is exploited in diodes optimised for the effect and known as varicap diodes. They are typically used for tuning solid-state analog RF circuits.