So I decided that the name Sun Soldiers is used only for those who previously were a part of Apparat's Soldat Sol, who now, with their strange tattoos, religious zeal, and unnerving loyality to the Crown became sort of a special force, "an order within an order", while the rest of turned-Grisha, so dockworkers, villagers who lived close to the Fold, and those of oprychniki who decided to stay/didn't manage to run(/weren't killed?) are still called Sun Summoners
Because when Alina was describing it in RaR, it felt like the power got transferred to a whole lot of people, and yet, in KoS and RoW we hear nothing about Sun Summoners, we don't ever see them in action, not a single one of them even gets a name, and the only three times they're ulilized is when there is need for someone to keep an eye on the Darkling.
There's no way an ex-oprychnik would be assigned to such task; they would recognize him immediately, even with slightly different face, and without his powers.
Ordering a turned-Grisha, whose otkazatsya's life didn't have to require from them to have a strong political stance, and bigger interest in who and how governs the country to undertake a task of such secrecy would not be the most cautious move either; three years living as a Grisha should radicalize at least some of those people. Alina was proclaimed a Saint, and the King still publicly called her "a witch" repeatedly, so I doubt "descendants to Alina Starkova's power" would be treated with much less animosity than your average Grisha. Their views, that earlier didn't exactly had to be very favorable to Lantsovs' rule, would be reshaped by their experience as a Grisha in a world that hates them, stories that foreign Grisha forced convinced by Zoya to come to Ravka would bring with them, and whatever would or would not slip throught previously-Darkling's Grisha's and ex-optychniki's mouths (because they wouldn't be able to afford being caught critiquing the Crown in the slightest, and silence can be as telling as words). There's little to no chance they would recognize the Darkling, but they don't have the faith that would ensure they won't accidentaly say something about who and why they were ordered to keep the check on. So they also aren't the perfect choice to guard the ancient evil, who officially is the most skilled manipulator to ever manipulate.
And so, since we dont get to actually know nor see any Sun Summoner, and the one we hear talking was a part of Soldat Sol, and they are doing tasks that should not be entrusted to anyone who isn't blindly loyal to the Crown, and I dont like the name Sun Soldiers, I decided what I stated in the beginning.
And if somebody wanted to argue those Sun Summoners cooperating with Hringsa are called Sun Soldiers as well, and there's no chance that there is this many of ex-Soldat Sol among turned-Grisha - I don't care. They're dismantling a millenia old religion that should be woven into every single part of the society with idiotic, uselsess "miracles", I don't care.