Idea number 1 -- Those who sought teachers for their sons wanted them to develop the skill to gain the voluntary support of other men. What a profound step this was to overcome “might makes right” with logic in a way that would put man on a path to rational behavior. We owe the stable political systems of today to this concept, and, even though men fall back to the use of force from time to time, the world as a whole has accepted the legitimacy of a government based on wisdom instead of force.
The Sophists built a system of higher education in Athens around a concept of Greek culture different from the culture of non-Greeks. There was a sense of pride on the accomplishments of Athens and the power of the new Athenian Empire. Maybe this was overdone and more hubris than pride, but it was a natural result of the position Athens held at the time.