"Ground: assigns a gate to the ac and clears the ac to taxi to this gate along the designated route"
I cannot think of a ground controller who will actually assign a flight to a gate, usually the airline's operations will do all of the when the crew calls in range, or ACARS will give the crew that Information based on what gate is assigned to them from airline operations or the airport operations. The crew will tell the ground controller what gate he is going to, and the ground controller will either direct him to the gate, or just tell him to taxi to that gate.
"Clearance/Director will provide departure cleance and innital taxi clearance"
Unless the Clearance controller is working both ground and clearance, he or she will not provide initial taxi clearance. Like I said there are expect ions, like when it is late at LAX
the ground controller will also handle clearance, but usually the controller will have the aircraft switch to the different frequencies even though its the same person working the two positions.