I think it makes sense to call a BJT current controlled when you compare it to the MOSFET.
The MOSFET has a gate, and the higher the voltage on the gate (which draws essentially no current), the higher the conductance from drain->source. So, this is a voltage controlled device.
Alternatively,
A BJT has a base. If you pump current into the base, The higher the there is conductance from collector to emitter. If a BJT had an alpha of '1', then no current would be required in the base. But, no BJT has an alpha of '1', so any real BJT requires ahigher the base current. You cannot power a BJT without current.
As a practical example which really highlights the difference:
- Flash Memory
This memory topology is impossible to implement with BJT's, because a constant base current is required for conduction. In a MOSFET, charges can be injected into an insulated gate. If they are injected, they will stay there, and keep the MOSFET conducting all the time. This conductance (or lack thereof, if no charges were injected) is sensed, and used to read the stored bit-state.