When boiling water without any other ingredients in it, why start from cold?

There have been plumbing systems in which the hot water was likely to have dissolved more [toxic|unsightly|unpleasant-tasting] material from the pipe walls or joints than the cold. In particular any system that uses lead-based solder, can leach minute{*}, but detectable amounts of lead into the drinking water, and the hot water is more efficient at this.

In this case that advice amounts to "use the clean water".

{*} Really minute. Like "Use this water all you life and not suffer any ill-effects" minute. But it can be detected, and who wants to chug down a glass of lead solution...

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."

My hot water tap is supplied by a combi boiler which heats the water on-demand, supplied by the same cold water source as my cold water tap. As a result, I'm confident that the water is reasonably fresh and clean. I don't use it for brewing tea or coffee, but I'm happy to boil vegetables and rice in it. It saves a couple of minutes bringing the water to the boil.

My parents' hot tap is supplied from an insulated immersion heater tank. The same water can sit in there for days, and it may heat and cool several times in that period. It is supplied by a header cistern in the attic. Last time I looked at the header cistern, there was a crop of dead flies floating on the surface, and some unidentifiable gunge settled at the bottom. This hot water is suitable for bathing and cleaning; it's not suitable for cooking.

If you don't know the details of your plumbing, and you're not sure it's safe, don't cook with water from your hot tap.

There has been a lot of discussion over the years in cooking circles about whether cold water or hot water comes to a boil faster, and the people that wrote those recipes are passing on their determination on to you. There is zero functional difference between one pot of boiling water and another. Once a roiling boil has been achieved, the water will always be the exact same temperature. That is one of the reasons that we use water as a cooking medium.

No, there are no different physical processes. The only thing to be wary of is that water heaters can be pretty icky places so, depending on your system, you might want to avoid drinking that water. On the other hand, if you have an electric kettle then by all means use that to heat the water: it's clean and, like your water heater, much more efficient than heating water on the stove.

The idea that cold water boils faster than hot water is complete and utter nonsense. Suppose you have a pan containing a litre of water at 40°C. That pan will take some amount of time to boil – let's say five minutes. Water has no "memory" of its past. It does not know how it got to 40°C so how it got to 40°C cannot possibly influence how long it takes to boil from that point. In other words, any litre of water at 40°C will take five minutes to boil in that pan. In particular, if you put a litre of water at 20°C in the pan and start heating it, it will take some amount of time to come to 40°C. After that, it will take the standard five minutes to boil. In other words, surprise!, cold water takes longer to boil than hot water because, first, the cold water has to be turned into hot water and then the hot water has to be boiled.

The specific heat capacity (SHC) of water is 4.184, which is a measurement of how much energy (in kilojoules) it takes to raise 1 liter of water by 1 degree celsius. Your stovetop is constantly generating heat energy and transferring it through the pot and into the water, which absorbs it and raises its temperature. Once the temperature reaches 100 degrees celsius (212f), the water begins absorbing that heat energy to convert to steam, which converts to a gas at the bottom of the pot and rises to the surface, causing boiling.

So let's say the average recipe uses 4L of water. Tapwater comes out around 4 degrees C, and your water heater has water at about 60 degrees C. In both cases, the temperature will have to be raised to 100 degrees C, and while we don't know how much heat our stove generates, we know that it will be the same.

The amount of heat required to raise an amount of water by a certain temperature can be shown with the formula:

Q = mc(T2-T1)

Where Q is the total energy required, m is the mass(4L of water = 4kg), c is the SHC of water (4.184), and T2 and T1 being the difference in temperature. So if we plug in our respective numbers:

Q = (4kg)(4.184)(100-60) = 669.44 kJ

Q = (4kg)(4.184)(100-4) = 1606.656 kJ

So as you can see, the cold water requires 2.5x more energy than the hot water. Your stove will be outputting a certain number of kilojoules per second of heat energy and once it has put in the required amount the water will reach 100 and start to boil. These numbers may vary based on different temperatures of hot and cold water, but warmer water will always boil faster.

My high-school chemistry teacher claimed that hot water is inferior at storing dissolved gasses, and that hot water has probably been sitting in the plumbing system for some time, so a great deal of the dissolved oxygen gas has been expelled. On the other hand, cold water from the taps is more "fresh" and enriched with oxygen gas. Even after boiling water, the trapped gases will take some time before they escape. So if you start with cold water, whatever you're cooking with the water will become more enriched with oxygen gas than if you started with warm water.

Dissolved oxygen gas = tasty?