Eggs – Does it matter if I start boiling hard eggs in hot or cold water

Some people say cold water boils faster than hot water, this is false, found here and here.

One reason might be (from the first link): "Some water heaters may introduce additional sediment into the water, giving you another reason to consider starting with cold—at least, if time is not of the essence."

The most obvious thing is nothing to do with heat/temperature. The rapid boil agitates the food a lot, to the point that if the food is soft, it can pretty much tear it apart. You probably don't really want disintegrated food, but smaller pieces do cook faster, so I suppose you can look at this as a rapid boil cooking faster from a certain perspective. It certainly cooks differently.

The other big thing is that rapidly boiling water will recover faster when you add food to it. It's not because the water itself is hotter, but the pot is, and the stove is too if it's electric. So you want a rapid boil to start, even if you don't need it later.

As for heat/temperature, there are differences between a full rolling boil and a slow boil, but not really what you're suggesting.

At a rolling boil, the water is mixing well enough that effectively it's all at 100°C. At a slow boil, it's really only boiling at the bottom, with a few little bubbles floating up from there, so most of the water is actually a bit below 100°C.

This difference is bigger than any effect from steam coming into contact with food; the heat capacity of water is substantially greater than that of steam, the steam isn't under pressure so it won't be above 100°C, and the food will be in contact with water more of the time than steam anyway.

Of course, if all you're doing is boiling water with a relatively small bit of food in it, it doesn't make a huge difference if the water's a little below boiling. But if you've got a lot of food and not that much water, like in a stew, the difference can become way more pronounced. Convection becomes inefficient, so at a simmer or a low boil the heat doesn't propagate from the bottom to the top very efficiently. That lets the temperature at the top be significantly lower, and so things will cook more slowly. Covering the pot does mostly mitigate this, if it's an option.

Finally, the bottom of the pan is substantially hotter than the water, and if you've turned the stove up higher to make it boil faster, it'll be even hotter, so food that comes into contact with it will cook (or more likely, scorch) faster. That's not directly due to the faster boil, of course, just the heat being transferred from the stove to the pot, but they go hand-in-hand.

So yes, things do sometimes cook faster at a rolling boil (what you call "boiling it really hard") than at a slow boil, but it's not because of steam coming into contact with the food, and once it's truly boiling, adding even more heat doesn't really change anything in terms of heat.