Years of poaching eggs in a pan of water tell me that cooks don't always know best.
The thing that cooks your egg is the temperature of the surrounding water. If you use a small pan of shallow water, the total water volume isn't much, so when you add an egg that is colder than the water, it will remove heat from the water, and the temperature will drop to a point where it is not hot enough to cook the egg. To combat this effect, they need to keep adding heat to the pan of water - hence the need to simmer. This slight boil causes movement in the water that can break up the white of an egg which isn't very fresh (that bit I do agree on).
My solution is use a deep pan, with a large amount of water in, vinegar is not required. That way, when you add an egg, the temperature will still be high enough to start to cook the outer surface of the egg.
Bring the pan to a good boil, then remove from the heat. Crack your eggs into the pan (Use a big spoon to lower them if it makes you feel more in control) and leave off the heat until he outer layer of the egg starts to cook. It will form a pale white "skin" which will then hold the rest of the egg safe. Put the pan back on the heat and warm the pan to the point where the water just starts to boil again, then turn heat down to keep it at that point. At this point the inner bits of the egg will be cooking, but if the pan starts to boil too much, the "skin" of cooked egg will protect the inside from the violence of the bubbles.
Test your egg with a blunt "butter" knife to find if the white is firm, or still not quite cooked. When it is the way you like it, lift out with a slotted spoon (plastic ones are thicker than metal, and more blunt, so less likely to cut into an egg), avoiding any loose stringy bits of white from your week old egg.
I quite like mine with cottage cheese straight from the fridge. The contrast of the warm egg and cold cheese is one I find pleasing.
Geoff the Duck.