Was Caesar a king?

Someone led a Christmas quiz recently. Most of the answers were "The Bible doesn't say."

But one question was how many kings are mentioned in the Bible accounts of Jesus' birth, trying, I suppose, to see if we'll count the Magi or not.

The "right" answer is 2: Herod (the Great) and Jesus.

I insisted that Caesar Augustus was a king. "No, he was an emperor."

When I said that an emperor is a kind of king (a king over kings), I was told that that was "semantics".

I don't know what people mean when they say that to me. I'm not sure that they do either. What I hear (correctly or not) is "that's thinking too hard about what words mean, their various categories, issues of translation etc." as if that is a bad thing.

John 19:15 ("We have no king but Caesar.") didn't appear to convince him or her either.

So, what do you think? Was Caesar Augustus a king?
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I understand a "king" (and its equivalents in other languages), as opposed to a Duke, Earl etc., to mean a sovereign, someone who answers only to God (theoretically, but allowing for modern consitutional monarchies).

HOWEVER, there are problems with that def'n if a king is ruled by an emperor. In that case, the emperor would be the only king. The vassals kings are more like former-kings, but (as with Herod) they are indeed called kings.

As well, I think there have been duchies that have considered themselves sovereign (and were, as the word implies, ruled by a Duke).
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BTW if "the Christmas story" goes all the way until the end of Matt. 2, then Archelaus is mentioned as succeeding his father Herod.
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And was Caesar Lutheran?