When I say the flight was on time, it means it was on schedule to leave at that specific time. They made it in time for departure means they had enough time before the departure time.
It will happen before a given time in the future. Enough time.
Ex: I arrived just in time for the meeting. (Arrived before the meeting started)
Ex: He came home in time for dinner today. (Came home before dinner)
Ex: I got home in time to watch the basketball game.
It will happen exactly at the fixed, planned time. Not too late or too early.
Ex. Is the train on time today? (Not late or early)
Ex: Your presentation is tomorrow morning, so please be on time. (Arrive at the class time)
Ex: I try to get to work on time everyday so I can leave on time. (At the planned time)