Abuse of Authority does not, for instance, cover "I'm the boss, I'm spending all of our funds on cocaine! Hahaha!" or "Hey, I'm going to the brothel, you do the paperwork for me."
It basically just covers "abusing your subordinates too much" (where "too much" depends on who you ask, but a good guideline would be "interferes with their regular duties and studies").
"Failing those under your duty of care" is pretty similar, though it more-or-less means "letting bullying and such happen underneath you". In other words, if you have authority over someone, you should not let them get away with serious offences.
As for the headmistress:
Silver Wing Tyrant (Solar: Zenith) is fairly old, and reasonably well-known across the realms as a philosopher and scholar. She is not known to be an accomplished fighter (though probably has some skill just by dint of surviving this long, and she *does* tend to carry a goremaul that can change to a cane as she wills it). Most of her time of late has been spent in Yu-Shan, pursuing scholarly matters.
She wants this to succeed, and indeed wants to turn the school into an army of those ready to change the world and bring about the new era. That said, while she doesn't concern herself with every single deed, she is swift to deliver punishment for anything she personally witnesses, and odds are good that she enjoys it. Basically, Whisper might find an appointment with the headmistress only if her actions are soul-crushingly cruel.