When you say "interesting interpretation", which of the numerous things I mention are you talking about?
When I mentioned energy, I was talking about energy delivered to the customers. 'Consumed' in the layman's terms but converted, as I mentioned, in more scientific terms.
You used the term 'power to be produced'. That terminology is where I have the problem. I would say energy is produced (converted) at a certain rate. You can think of 'using' power, but the only way you can purchase it is by measuring how long you use it. That is why the electric bill mentions the number of kWh you use. They could send you a graph of the instantaneous power versus time and show you the area under the curve, along with you peak power, but they can't charge you for power alone. Instantaneous power is interesting since we can 'see' what is going on and it does measure the stress on the delivery infrastructure, but energy delivered, (the area under the curve) is what we buy.
As far as being more impressed by current or current density, since current density is current divided by area, there are several ways to increase it. If we are looking at the main transmission line alone, the area of it will remain roughly constant, so quadrupling the current will quadruple the current density. I am equally impressed.
When someone proclaims a new power record, the public does not realize what it means. They don't realize that that record may have existed only for a few minutes. It does not necessarily mean record energy production (conversion) for the day. That is where the media lack of understanding distorts reality.
I am glad Maxwell's equations are helpful to you but they don't help the average guy trying to figure out the difference between power and energy.
The current travels on the surface of the conductor rather than the cross-sectional area. Current density is closer to the measurable phenomenon. You might enjoy of good series in the Philosophy of science, studying the etymology of the terms used to express physical phenomenon and their historical progress in differential math.
The Poynting Vector might provide better insight, after one memorizes Maxwell's Eqns in differential and integral format.