General classifications of amplifiers
You missed the 4th type:


Trans-impedance amplifier -> Current-controlled Voltage-source


Maybe it is not in your list since devices behaving like that are not so common.

Anyway, these are all 2-port devices.

An ideal 2-port device which is voltage controlled should have an infinite input impedance so that it will just "sense" the voltage applied across it.

An ideal 2-port device which is current controlled should have an zero input impedance so that it will just "sense" the current flowing through it.

Similar (but impedances reversed) for the outputs:

An ideal 2-port device which has a voltage output should have a zero output impedance, any load current will not influence the output voltage.

An ideal 2-port device which has a current output should have an infinite output impedance, any voltage applied at the output does not affect the output current.

Now relate that to the physical devices you mention like BJT or FET. Since these are real devices (and not ideal) their input and output behavior does not exactly fit any of the 4 2-ports.

It is more or less a matter of how you want to use the device (in which circuit configuration) that determines which 2-port model fits best.

Since a FET has a high input impedance I would not classify it as having a current input. A BJT's input impedance can be quite high (Common collector circuit) or very low (Common Base) or something in between (Common Emitter). So I would say that for a BJT it depends how you use it.

The well known Hybrid-Pi small signal model of a BJT is based on a transconductance (voltage to current) but that model can be used to make and evaluate all three BJT configurations. I'm sure that if you made a current to current based small signal model for a BJT the same could be done and the results would be the same.