I am trying to grasp the difference between voltage, current, power amplifiers... I understand the gain in all three scenarios (the ratio between output and input quantities, V_out/V_in, I_out/I_in and P_out/ P_in).

For example, it is possible to build a circuit using BJT transistors (which I consider current amplifiers) that works like a current amplifier but also like a voltage amplifier. What is the difference? Does it depend on the input and output impedances of the amplifier circuit if we call it voltage or current amplifier? For example, if the input impedance is large, will the input be a voltage? And if an amplifier has low output impedance, will generate a larger output current... I am confused on these concepts....

Power is P=IV so to amplify power we can either amplify both I and V or either one while keeping the other variable constant.

There are also devices that have an input current and amplify the output voltage...