As many others have pointed out, the system itself isn't going to change a lot outside of large vehicles that would be able to be converted to pure electric power as they could carry their own box.
However, there is something that does not seem to be touched on in the question and is very important when dealing with electrical power. How has battery technology advanced and/or been affected during the time leading up to this discovery? I consider this to be the real tipping point here because of the issue being raised about peak power. Unlike some sources of power, electrical energy can be stored to at least some extent. As such, if the battery technology has improved along the way to this cold fusion break through, then not every single person or household would require a box to themselves.
I'm not going to bother with the math, others can do that if desired, but assuming a decently capable battery, I would gather that the ability of the machines to be shared would increase greatly. 284
Instead of using them as direct energy feeds to a grid system, they would act more like trickle chargers working constantly to keep a system of batteries charging. The batteries would function as the actual source of power as far as households and the like would see, and during peak times they would drop power, but would ideally have a capacity to deal with that for some time and then recharge during periods of lower usage demand.
Efficient batteries would mean that the magic boxes could be more of a shared purchase, while the batteries would be the majority of the personal costs. Each household/business would have to decide how much power they need and can afford to store and make their battery/box purchases accordingly.
Or power companies would be the ones buying the boxes/batteries and continuing to dole out power to each customer as needed, charging in a manner similar to cell phone companies where you would have a certain amount that you were entitled to based on your payment plan, then extra fees if you go over. They would just purchase more boxes and batteries as demand for services rises. I consider this to be the most likely outcome as it would require the least amount of change to implement. 284