Sylvanas was mind controlled just like the entire Scourge except perhaps members of the Cult of the Damned (necromancers, Acolytes).
Arthas considers people of the scourge as his subjects and they consider Arthas as their King (however Ner'Zhul is still the big boss).
Therefore indeed, Sylvanas betrayed Arthas when she broke free. This is why Balnazzar is saying : "The lich, Kel'Thuzad, is far too loyal to betray his master. But you, on the other hand..."
https://wow.gamepedia.com/A_Kingdom_...d_(WC3_Undead).
Note that the details about the mind control were always vague : For exemple Arthas never directly mind controlled Sylvanas. It was Ner'zhul. It is perhaps why he didn't know that she broke free.
Sylvanas Windrunner: "How could I not?
For some reason I no longer hear the Lich King's voice in my head. My will is my own once again."
https://wow.gamepedia.com/Sylvanas%2...l_(WC3_Undead)
Here it's implied that hearing the Lich King's voice = being under mind control.
This is why lot of people think that Arthas was just a puppet of Ner'zhul, like any member of the Scourge, having no free will since he took frostmourne. Hence why Malganis thought that he won when Arthas picked the sword :
Arthas: "You waste your breath, Mal'Ganis.
I heed only the voice of Frostmourne now."
Mal'Ganis: "You
hear the voice of the Dark Lord. He
whispers to you through the blade you wield. What does he say, young human? What does the Dark Lord of the Dead tell you now?"
https://wow.gamepedia.com/Frostmourne_(WC3_Human)