I think you are looking at batteries (which store energy (Volts * Current * Time)), rather then a load which consumes power (Volts * Current).
A Battery is effectively rated in stored energy (Kind of) hence units of Ampere Hours, where a load is rated in power dissipated, hence just amps (For a given voltage in both cases).
This is useful as it means that by dimensional analysis you can simply divide a battery capacity (Say 100 Amp Hours) by the total load current (say 5A) and get the run time, 20 Hours.
Now current is of course rate of change of charge, and 1Ampere is defined as 1 Couloumb per second, but I doubt that is the definition you are seeking.
Regards, Dan.