First day at school. The principal is giving a long and winding and really dull speech.
A student tells a pretty girl in the crowd, ‘Man, that’s some dull speech, huh?’
The girl says, ‘Hey, do you know who I am?’
‘The principal’s daughter.’
‘Oh really? You know who I am?’
Διευθυντής = principal, only in schools. For universities, διευθυντής would refer to a director (of a department), not a specific faculty position.
Μαθητής is only used for school (up to and including high school). Then you have φοιτητής as the most common and σπουδαστής (I think this is reserved for lower tertiary education).