A common misconception is that Satori (the Lord of the Mountain) appears randomly and that the player doesn’t have influence over whether or not the mountaintop will glow. While a little bit of RNG is indeed involved in the process, not all of it is random:

The LotM appears during a random hour of the day/night of the waxing crescent^[1]. The in-game moon phases follow a strict pattern (independently from the Blood Moon BTW) so you can just skip a couple of days at a campfire to get the desired result. An easy way to check for the current moon phase is by talking to Hino at the Dueling Peaks Stable, provided you don’t want to wait for nightfall to check the moon phase yourself.

Once you fast-forwarded to the correct day/night, you need to let time pass naturally - no more campfire time skipping for now.

You need to stay far enough away from Mount Satori that you’re not on the same map unit, but close enough that you’ll notice the green glow once it appears. The IMHO easiest place to wait is atop one of the Sheikah Towers as you’ll be able to see the mountaintop from most of them, and enemies won’t bother you while you wait (no keese / stals up there).

Once the mountaintop glows in that northern-lights-like sheen, it is safe to warp to the mountain (or run/ride there if you’re close enough for that). You’ll be greeted with an eerie music, blueish-green haze and a ton of Blupees gathered around the cherry blossom tree, where the LotM is either already present or will appear shortly after. He sometimes disappears briefly, but will reappear as long as the eerie music is still playing. If you scare him away, he’ll despawn for real, the music will fade and the haze will vanish, and then you’ll have to wait for the next crescent moon before he appears again.

Why would you want to “summon” Satori?

He can be caught and tamed, but not registered, so forget about keeping him as a permanent mount. And unlike Blupees, he won’t drop any rupees (or other loot for that matter) if you hit/shoot him, so you won’t gain anything by attacking him.

He does however have an unique Hyrule Compendium entry, so if you’re interested in reading up on the in-game lore or just want to complete your compendium collection with personal pictures, you need to encounter him at least once during the playthrough.

A common misconception is that he allegedly has perfect stats (5/5/5) but datamining has disproven this claim: he is actually “just” a 1/3/5 mount with 100 HP and the unique feature that his spurs regenerate immediatly so you can make him sprint infinitely without a penalty. As a tradeoff, he is prone to just disappearing if you dismount for even a couple of seconds, so don’t leave him unattended for too long.

(All credit for that map goes to HylianAngel)

As for the mountain itself, that’s where things get interesting, because even tho the mountain already has a ton of neat, rare materials availiable (Endura Carrots and Silent Princesses for example), there will be even MORE rare items/animals (like an entire tree full of beetles) and a flock of Blupees present during that time. Plus, all Blupees across the rest of the map respawn and stay until you scare them away or hit/shoot them, making it possible to farm them for rupees again. Also, the stal-monsters normally present on the mountain simply won’t wake up.

PS: The game does check for whether or not the LotM already exists, meaning that you can not spawn a second one while you’re still riding the one you formerly caught. You have to release and despawn the first one, otherwise the mountain simply won’t glow.