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The name comes from the fact, that there is a more or less fixed ratio between output and input current (Ic/Ib=hfe), thus the BJT can be seen as a current controlled device. But in physical reality it is NOT current controlled.
The correct wording would be "voltage controlled" or even more exact "load controlled", because the input current at the base pin does not control anything.
In contrast, it is a parasitic current.
I think it is wrong. The conduction mechanism in BJT is based on two current components, hole current and electron current. If there is no base current, there will be no collector current. The base current is used to change the resistance in base region (which is very thin) and then to control the larger collector current. In contrast, the conduction of FET can be controlled with Vgs only.
Sorry, but the mechanism behind current control is slightly different.
For simplicity I neglect the influence of the holes, OK ?
At first, the base-emitter voltage causes the emitter to inject electrons into the base region. Because of their kinetic energy (velocity !) and the collector voltage most of them can "jump" over the "fence" (that means: can move through the base region) and are collected from the collector. Only "a few" electrons don´t reach the collector and make the so called base current. But these few electrons do not "control" the main current in any way. However, they constitute a more or less constant percentage of the collector current - and therefore the simplified (but in reality useful) sight that the BJT is "current controled".
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