Is this basic schema of a brushless driver correct?
Yes it is basically correct.  The connection to the motor could be more clear, which may be why you are confused about where the current goes.  I found a nice picture on the web that's a schematic representation of delta and Y connections.  With a brushless (or any 3-phase device), you're always driving at least two terminals at different voltages, so there's always current being driven in at least two coils.

In a typical BLDC setup, two terminals would be driven, one high and one low, and the third would be floating.  In a sensorless motor, that third terminal would have its voltage monitored, which would be used for commutation.  In a sensored motor, the shaft position relative to the phases is read directly.  In a BLAC all three terminals are driven, and if the motor is sensorless the current consumed in each leg of the driver is used to determine commutation.