Re: So just what does 'charges in minutes' mean?
"Quick charging" claims are always bogus.
The only way to manage that is to have incredibly high voltages involved, which become a danger themselves as that allows them to arc more easily. So you'd need something like a 400V battery to make it sensible, and step-down circuitry in a mobile phone to get it back to sensible levels.
And the batteries we use everyday, worldwide, tend to be thoroughly in the stupidly-low-voltage ranges. 12v is common, 24v is around but less common, anything higher is considered "specialist" and usually purely to avoid high currents as you specify (most larger home solar installations are 24v purely because it then halves the current so you can still use the same cables/charge controllers without having to upgrade - in some models, the charge controller and cables are identical between kits and you just change the batteries for 24v batteries as you expand the system capacity).
Your house probably only has 100A incoming at most. You're basically saying that you're going to deliver enough current - for a short time - to power your entire house. You're talking bomb-level energies here if that goes wrong.
There's a reason that even electric car chargers are still basically 240v (or even 400v) and HOURS of charging. I have a 32A commando connector (building site 220v connector) on the side of my house. It powers a 20A electric kiln to 1600 degrees. It's a scary amount of energy, and a scarily thick cable to do so safely, and cable thickness is related only to current, not voltage - I^2 * R. I could charge an electric car with it and it would still take hours. And, yes, that's 240v but the reason we use 240v is PURELY to decrease the current-carrying-copper-thickness required for currents (at 12v AC with the same power, we'd be pushing 20 times the current, up to 260A, just for a normal 13A appliance - and 260A requires something ludicrous like a 30mm-thick conductor for each part of the cable).
What they are suggesting is that your mobile phone charger would somehow build up that amount of current (I assume it would build it up over time and then deliver it in one quick charge, like electric welding kit, as otherwise it would just fuse your house) down a tiny cable into a tiny battery, and get the charging time EXACTLY right and be able to detect faults quick enough to stop before the battery blows.
It's ALL a nonsense.
Quick charging won't happen until we have high-voltage batteries, chargers and convertors small enough to put in the devices in question. We haven't bothered to do that for electric cars yet, so fast-charging of mobile phones or AA's is still a nonsense. There's a reason your laptop batteries are 19v, because that keeps the amps down (2-3 in your average laptop?). At 12v, the same power would draw up to 5A, which needs thicker conductors throughout.
Increased power = increased current or increased voltage.
increased current = thicker cables
increased voltages = more difficult voltage step-downs, less common hardware, greater arcing distance, etc.
There's no way around it.