I understand the difference between voltage amplifier and power or current amplifier. For example in an audio amplifier there are usually three stages: two voltage amplifiers and a power amplifier at the last stage. [...]
[emphasis mine, N.A]
What you have described is the difference between signal amplifier and power amplifier. The signal happens to be voltage and power happens to be current in this case.
Consider this. There is no general definition of gain, or even units of measurement for gain. The gain and its units are defined on case-by-case basis.
When you wish to drive coils in the speakers, you care about output current, while voltage is less important as long as you have enough compliance voltage. The gain1 of the current output power stage stage would be defined as Iout/Vin, and units would be [A/V].
When you wish to drive piezos (for a different example), then you care about output voltage, while current takes the second seat. The gain1 of the currentvoltage output power stage stage would be defined as Vout/Vin, and units would be [V/V].
1 I'm assuming that the signal is voltage (perhaps provided by an upstream signal amplifier).