I was doing some irrelevant philosophical research when I came upon a text saying that the Ancient Greeks had no proper word for "authority". I thought the best place would be to ask here since I always welcome additional historical context. Did the word "democracy" also stand in for what we could call "authority of the people", and because they had a word for everything related to authority there was no need for "authority" as such?
I'll reject the premise. Ancient Greek has plenty of words for 'authority'.
'Authority' has a bunch of senses -- permission to perform a role; command over others; legal autonomy; trustworthiness as a witness or source -- so it'd be bizarre if there were no word for any of these concepts. 'Lawful authority over policy' or 'responsibility for a role' could be ἐξουσία. 'Respect, influence' could be δόξα or ἀξίωμα or a bunch of other words. 'Rule, command in government' could be ἀρχή. And so on.
If the text you saw stated that there was 'no proper word for authority' on the grounds that there's no word that means all of these things simultaneously, then the author is just being silly. It isn't normal for a given word in one language to have a complete and exhaustive equivalent in other languages. That's the kind of thinking that leads to nonsense like 'ancient Greek had no word for blue'.
Words in -κρατία, like 'democracy', refer to the source of governmental authority, not the authority itself. They specify a type of government, not government in general. So 'democracy' specifies a government where the origin of legislative power and governmental authority rests in the people, as opposed to the origin resting in some more limited group.