Beyond mathematics, and semantics, the important part to me would be to think of it in terms of what unit everyday items are rated in. Each type of compenent is rated by a different unit, because that unit of measure best explains what you are trying to do with the items.
For example:
Voltage: This unit is used for current conductors (wires.) Wires are rated in voltage because high voltages require expensive insulation. Also; voltage drop, is based on wire size. That is; high voltage can use smaller wire size, but will need expensive insulation.
Current: Most load devises are rated in "amps." This is because most devices are designed to draw a certain number of amps. Increasing the voltage will increase the horsepower (or watts). Many motors are controlled by VFD (Variable frequency drives, which adjust the 60hz cycle or the voltage. this is also true of light bulbs. So in most loads the work (watts) can be variable, so can the voltage, but typically the amperage (current) is constant.
Power: Watts is the preferred unit of measure for generators. This is important because watts measures the amount of work that can be performed. A generator can use a transformer to change the voltage or the amperage to virtually anything. Wattage is the only thing that is constant, and consequently the most important unit because it measures the amount of work that can be done (e.g. the number of light bulbs, houses, air conditioners, etc. that can be run.)