> between the current (as in BJT) and the voltage (as in field transistor and valves)
The BJT is also much more accurately modeled as a device where the collector current is determined by the input voltage (i.e. base-emitter voltage), exactly like the vacuum tubes and the field-effect transistors.
There are however a few reasons why their description as being controlled by the input current is preferred in popular literature.
One is that the dependence between output current and input voltage is more strongly nonlinear than for vacuum tubes and FETs, i.e. it is exponential instead of being polynomial, so it can be approximated as linear only for very small voltage differences, of a few millivolt.
On the other hand there exists a relatively small current range where the collector current depends linearly on the base current, which is useful for mental approximate computations of a circuit with BJTs. But it must be kept in mind that this approximation is not useful for accurate circuit design, because beta (the ratio between the collector current and the base current) instead of being a constant it drops quickly both at big collector currents and at small collector currents.
Besides the great non-linearity of the output current/input voltage dependence and the approximate linearity of the output current/input current dependence, the other reason why BJTs are frequently described as "current-controlled", instead of the more appropriate "voltage-controlled", is that their input has a great leakage current (i.e. the base current), while the leakage currents of the inputs of vacuum tubes and FETs are so small that they are normally negligible.
In conclusion, one may prefer to use the description of the BJTs as "current controlled", but one must be aware of the serious limitations of this point of view. Even for mental computations, it is more useful to use the beta value of a BJT not for computing a fictive current gain, but to compute the input leakage current of the BJT conceived as a transconductance amplifier (unlike beta, which varies from transistor to transistor, the transconductance gain is the same for all BJTs), to be used for computing voltage drops in input resistor networks.
The animated visualizations are nice, but the only way to easily assess the quantitative differences between various types of BJTs, FETs and vacuum tubes is to draw the families of curves that show the output current from output voltage dependence, having as family parameter the input voltage (and also the family of curves obtained by interchanging the output voltage and the input voltage between graph axis and curve family parameter). (These families of curves being sections of the 3D surface of the graph of the output current as a function of input voltage and output voltage, but a 3D graph does not allow a precise comparison of the values in different points.)