Thanks "Crazy". But on one egg the shell actually had a dent in it, so I was afraid that the white would leak out before it was cooked. I've had that happen.
I need a "machine" that puts a pinhole in an egg. I know I use to have a little thing like that, but I think I gave it away. It was a small plastic piece that just cradled the end of an egg. In the center of the bottom there was a tiny needle point sticking up about 1/8 of an inch. Now I'll have to go out and buy another.
I did steam these eggs the waterless way. Put some paper towel or a small cloth on the bottom of the pan, add just enough water to be absorbed by the cloth. Pour out any excess. Then put in the eggs and bring them almost to a boil; turn the heat way down and let them steam for 20 minutes. Makes the eggs very tender with no dark ring. I'll just have to learn how to peel them.

I'm another one that uses an egg cooker and the eggs are always easy to peel. My SIL tells me all these tricks she tries and then always has trouble peeling them. My mother always said she only had trouble when she is making the eggs for company. The main reason I have an egg cooker because I have forgotten them cooking and lets just say that pan was never the same and I thought the smell would never leave the house.

This works for me. Boil and let eggs cool the when ready to shell lay one on the cutting board and cut it in half like you would for deviled eggs. After cutting slip each side out of the shell. Really works.

I have not tried all of your methods but find that when my eggs are done "cooking", I immediately pour off hot water and fill the pot with cold water adding ice cubes to make the water really cold. Keep up with the ice cubes for about 10 minutes or so...until I am ready to peel and I usually do not have any issues.

I put my eggs into cold water, put in salt, bring to a boil and boil them about 20 minutes. My eggs are boiled eggs. I immediately put in cold water from the tap. Let them sit awhile, crack shells, put into fresh cold water and let sit until cold. I crack real good, roll them on a surface and peel. I have done this way for at least 54 years.

I'm late reading this thread; however, putting a dash of cream of tartar in the pan before cooking boiled eggs makes peeling much easier. A dash of cream of tartar also works well in cooked beans to minimize their bad side effect. :-O