Ohm’s law alone does not apply to power. If you put a 4-ohm resistor across a 9-volt battery, does that mean the resistor is consuming (9*9/4) 20 watts? No. That would mean the battery would have to accommodate the resistor with slightly more than 2 amps, but it the most it can do about half an amp. So the answer is a high current amplifier is one where it stores enough energy to maintain voltage over the speaker’s varying impedance.
You cannot tell from the specs how much energy it stores because most of the manufacturers specify the power at only one impedance (an 8-ohm resistive load). A speaker presents an inductive load which varies with frequency. But you can get an indication of the performance if the manufacturer provides power ratings at less than 8-ohms. Levinson, Krell and the like maintain their voltage (energy) down to two ohms. If an amp can do that at a 1-ohm load, then it’s truly a high current amplifier. Literally.