Going to try to keep this post as simple as possible, to give everyone an easy understanding as to what the “High King” title actually means.
The High King title is a title given to a primary faction leader, when he/she is appointed as Supreme Commander of the Alliance forces.
This is why Varian was a High King, but Terenas was not. Terenas filled the same role as Varian did, as the figurehead, face, center, and speaker of the Alliance. As Lordaeron, and Stormwind were both the centers of the Alliance at their given times, and both of the two respective faction leaders took on the role of speaker.
The difference between Terenas and Varian though, was that Varian was a military commander as well, and an active combatant. Whereas Teneras was more akin to that of a politician. And so, Terenas, and the rest of the Alliance, appointed Lothar, and then later Turalyon, to command in their stead, as Supreme Commanders.
With Varian actively leading the military himself, and with him already being the face of the Alliance, he was appointed as “High King”. Which just means that he is a soverign faction leader within the Alliance, as well is the Supreme Commander of the Alliance’s military. Not that he’s above the other leaders of the Alliance. Just that he’s commanding the war effort.
Anduin tried to take up that role after his father’s passing, as shown by him personally leading the Battle for Lordaeron. So for a time, Anduin filled the shoes. But, possibly due to how badly the Battle for Lordaeron would have gone, without the unexpected aid of Jaina, and Anduin’s own realization of his lack of age and experience, Anduin stepped aside and instead others led the war effort, for the rest of BFA. And Anduin granting Turalyon the title, once again, of Supreme Commander, is Anduin officially stepping down from the position, and handing it over to Turalyon.
However, Turalyon is not a soverign leader of the Alliance, like Anduin is. Even though he is technically the leader of the Lightforged Draenei, he’s a bit different from other faction leaders. As the Lightforged Draenei are more akin to being a military branch of the Draenei, as opposed to their own independent nation.
And so, because Turalyon is not a soverign leader of the Alliance, such as King, he was not granted the “High King” title, and is instead given “High Commander”.
So, in short: You have to be a soverign nation leader of the Alliance, and be given the rank of Supreme Commander of the Alliance Forces to be considered “High King”. This doesn’t put you above the other Kings and Queens, and such within the Alliance. It is just a way to signify that you are the Supreme Commander of the military.
And if you are not a soverign nation leader, and you are granted the rank of Supreme Commander, then you are considered “High Commander”. Which is the exact same as “High King”, except that you personally don’t rule a nation, like a “High King” would, such as Varian and his nation of Stormwind.
On a side note: Previously Turalyon’s rank was called “High General”. It basically means the same thing as it does now. It was just going by a different name, as far as I understand.
Though it is possible this was specifically written to signify that he was commanding the army, since Daelin Proudmoore was commanding the Alliance’s navy. But by the point Turalyon had taken over, the second war was already closing. And even then, I don’t think he was officially granted “High General” until after the defeat of Warchief Orgrim Doomhammer.
In any case, he’s basically in the same role as he was before. Except maybe now he also might be giving direct orders to the navy as well, instead of indirectly/them acting on their own accord.