This is easy to do with children. I once convinced all three of my children and four of their friends, by speaking to them very quietly and calmly as they lay on the floor with their eyes closed, that I had glued their eyes shut, but that when I clapped my hands they would be able to open them. I let them squirm and exclaim for about 20 seconds and then clapped my hands and all their eyes popped open. They still marvel at it, but I've never tried it again in spite of repeated entreaties from my young grandchildren. Children HAVE to be suggestible, I expect, in order to learn. That's not to say it's an entirely good thing.
I recall a presenter on hypnosis saying that children are extremely easy to hypnotize and in a sense go around hypnotized all the time.
I don't recall doing anything of this genre when I was a kid, nor do I know of relevant search keywords. But, based on a read of the two linked pages, at minimum the first one looks very hypnosis-ish? Like, I'm pretty sure *I* could be hypnotized by something like that. The second one I'm not sure what to make of, though, given that it's apparently supposed to produce externally-visible skin-markings rather than just internal experiences; that is Off-Model for me in terms of what hypnosis should be able to do, so if that one works as described then that's indicative of some separate interesting thing going on, I think.
I definitely recall playing those kinds of games as a kid. I don't recall any really notable psychological effects; I think I mostly played them with my younger cousins rather than peers so maybe I wasn't taking them as seriously (or maybe I'm just less hypnotizable, who knows).
Huh, interesting! Some of this rings a bell as similar to things I'd read in a book (I think an old Klutz book about road trip activities for kids). I looked around and found at least people talking about having done similar things: https://www.reddit.com/r/tipofmytongue/comments/jkxdp3/tomt_childhood_sensory_games_crack_an_egg_on_your/ , http://www.fourthgradenothing.com/2010/07/round-ball-crack-egg-on-your-head.html
As a kid, the adults would hold my arms down as I tried to raise them as hard as I could. Then, after a while, they'd let go and my arms would rise "on their own" iirc. I don't know the answers to your questions, but it was a weird time.
I HAVE NO EXPLANATION.
I would move kids' arms around while their eyes were closed, and then their arms would raise on their own without the other child willing them to. It was also when I was ~10, and I have not done it since. Wow.
(And this was in Canada!)
We played a variation of what that link calls "Concentration" that was much shorter but did end in determining how you would die by what color you saw when your friend punched you in the back at the end of the rhyme.
We also had a different egg-on-your-head game that was supposed to just give you chills or ASMR or something by going through a chant and touching someone's back, shoulders, and head in certain ways. And there was a chant about "spiders crawling up your back" that has a similar effect.
I'm also reminded of "light as a feather, stiff as a board" but we never actually succeeded in getting the intended effect of that one. I'm not sure if it really "works" in any sense since I can't speak from experience.
I would call it inducing altered states of consciousness rather than hypnotism, though that's a bit of a mouthful and definitely falls under the same umbrella. What you describe (making yourself feel heavy, fucking with your proprioception, inducing psychosomatic paralysis) reminds me of the effects of a variety of types of meditation I've played with on my own or practiced when I was studying with a Wiccan coven. My old Wiccan friends would describe it as affecting your energy body using your conscious mind, though there's probably a materialist explanation. Maybe hypnosis/suggestibility really is the best descriptor.
My sister knew something similar to "Crack an egg on your head"/"Concentration" (part of it, anyway, the opening chant matches but the linked article is much more elaborate), but she did it to her friends, not me, so I don't know how it actually feels.