I know I'd seen that green light on Satori Mountain in my first nine playthroughs but I don't remember how often.
After watching Vinny's playthrough and saw him tame the Lord of the Mountain, I realized what that light was, and first time I saw it on my now-completed tenth playthrough I headed there right away, teleporting to the shrine. Three times I tried mounting the Lord of the Mountain but I ran out of stamina every time (even though Vinny did it with less stamina than me). After the third, he ran off and the light went away.
So, I loaded the save file from when I first saw the green light, which even showed the light in the image, but the light wasn't there. Nor was the Lord of the Mountain. Somehow the game remembered I'd seen him and carried that over to the save file so that I wouldn't get a second chance at him.
From that moment on, I went after Medoh, Naboris, and the Hebra and desert shrines, but every night I'd check Satori Mountain again. After beating Ganon, I re-loaded the save file and camped at the stable near Tabantha Bridge, which gives me a good view of Satori Mountain, I even stuck a pin there to remind me which direction it was. But after 276 days, he didn't show up again at all.
I got quite a few blood moons in between - though only one for the camping, on the second night. I figured the Lord of the Mountain might require me to travel and do stuff like the Blood Moon does, which I did in between some of the nights. I even rode my horse a bit, it got killed by a Lynel so after avenging it (figuring out an easy strategy for them in the process), I got Malanya to revive it.
But still nothing. I really think he only ever appears once per playthrough, or at least doesn't return after you try to mount him. Like the Divine Beasts, once you free the Divine Beasts and kill the Blight Ganons, that's a permanent change that can only be undone by starting a new game. Same with Kohga, who stays dead, and likewise after beating him, Yiga will randomly pop up throughout the land to get their revenge.
Has anyone here ever actually see him up close more than once on the same playthrough?