If Athens is legitimately a democracy then I am confused how you can come to the conclusion "one thing is for certain, a democracy can't be authoritarian by definition."
Athens killed Socrates using an authoritarian law after all.
I think we're using two different meaning of authoritarianism. It means both "undemocratic/rule of the few" (my statement) and tyrannical (your position).
Any state, democracy included, can be a tyrant (i.e. cruel and oppressive) against its perceived enemies.
Athens killed Socrates using an authoritarian law after all.