Anyway, of course it ended up being my mom's method that works for me every time. I don't know why I kept looking online when I should have just asked her. (In fact I DID use her method when I was living at home, but later I forgot and sought "cooking experts" online instead. Let that be a lesson to me.) I make no promises, because each of those other million methods made promises: "this works every single time," "this never fails," "the shells come off in huge elegant pieces," "the yolks are buttery-smooth"---and each one of them STILL didn't work. Which tells me there must be a bunch of other variables involved. But anyway, for anyone who is searching for how to cook hard-boiled eggs at high altitude (just in time for Easter!), this way works for me.
1. Put several eggs in a saucepan (try to pack them in fairly tight so they don't have too much room to roll around). Cover them with cold water, add some salt to the water, and put them on the stove over high heat.
2. Bring the water to a boil. I had a cooking teacher who made us learn to hear the difference between boiling and non-boiling water in a lidded pan (so we didn't have to lift the lid and let all the heat out). Anyway, stick around close so you can listen (or peek in) to know when the water starts boiling. As soon as it does, put a lid on the pan (if it's not on already), turn down the heat to low, and set a timer for 12 minutes.
3. When the timer rings, turn off the heat, take the pan off, pour out the boiling water in the sink, and re-cover the eggs with cold water. (Do this a few times, since the hot pan will heat the cold water pretty fast at first.) Let them sit in the cold water for 5 minutes or so. You can then refrigerate them for later, or peel them.
4. When you do peel the eggs, do it under cold running water. I think this helps break the bond between the shell and that under-membrane beneath (?).
UPDATE: Sam actually thinks that peeling them under WARM running water works better, and I think I agree. We've had more success with this lately, anyway.