Greek, Roman, and Elizabethan plays are typically either Comedies or Tragedies. But tragedies seem to have been replaced with the broader term of drama. When did this shift happen? Did it have to do with the rise of the radio and television and hence serialised stories?
I broadly define tragedy as: plays based around human suffering, in which most main characters come to a bad end. Modern drama can follow this format, but doesn't always; dramatic characters deal with problems which may or may not turn out badly. Tragedies seem to have been subsumed into the broader category of 'dramas'.
I broadly define tragedy as: plays based around human suffering, in which most main characters come to a bad end.
You more or less answered your own question. A tragedy is woven over a theme of inescapable fate, where one's actions inevitably lead to one's downfall. The story of Oedipus is probably the greatest, and most obvious example.
COMEDY, on the other hand is a play where despite dificoulties and danger, the protagonist reaches a satisfactory end (often, but not necessarily in a humorous way).
For simplicity's sake, it would be fair to say that prior to the Elisabethan Era of theatre, those two forms of drama were rather rigid.
As culture evolved, those two types of drama were blended, and mutated into various forms. It would be hard to pinpoint a certain date when "classical" tragedy and comedy becomes passe, but by the times of Shakespire and Molier, the new type of drama became popular enough for the two authors to gain fame.
Simply put, modern-ish (say, post XVII century) drama plot almost never follows the rigid rules of tragedy, and even "pure" comedy is rare. The vast majority of fimmakers and playwrights prefer to mix things up in a more nuanced way. Modern "dramas" rarely follow the structure of the tragedy anymore. This is because a true tragedy is wrapped around the idea of fate always taking its predestined turn (classical greek tragedy), or moral decisions always leading to "just" consequences (christian-european tragedy), and both of those worldviews are not particularily popular today.
sources:
Tarrant, R. J. - . "Senecan Drama and Its Antecedents".
Murray, J. - . Shakespeare and his Predecessors
Silk, M. - Tragedy and the Tragic, Greek Theatre and Beyond