"God's golden child" and the other names you mention are titles that are used in in this play to refer to Phoebus Apollo, the son of Zeus, Lord of the Gods in the Greek pantheon. Phoebus Apollo had a somewhat ambiguous role, as he was variously considered god of the sun, healing, prophecy, archery, medicine, the plague, music and the arts. His activity can be likewise seen as somewhat ambiguous. You might like to consider how he is invoked by the Chorus in the First Ode:
Phoebus Apollo, stretch the sun's bowstring,
That golden cord, until it sing for us,
Flashing arrows in heaven.
Note here how the Chorus associates archery with Apollo. The Chorus imagines Apollo both as a destroyer who can demolish the enemy of Thebes but also as a healer who can take away the plague.
For Greek's Zeus was their god, their immortal voice, their golden hope, and his son would be the child.