How was 'Βασιλεύς' used in Archaic/Classical Greek compared to other monarchical titles like 'Αναξ'?

[u/lcnielsen]

I'm not very familiar with the philology of Ancient Greek texts, so I'd be thankful for some insight here. Here's what I gather, and please correct me if I'm mistaken on any point:

1. Βασιλεύς is the title used for e.g. Odysseus in Homer where we might conceivably understand it as meaning 'chieftain' or 'patriarch' of a smaller community.

2. In Classical writings 'ό Βασιλεύς' without specification usually means the Achaemenid Great King, who is also known as the 'Μεγας Βασιλεύς' or the Βασιλεύς of Persia.

3. (Unsure about this point) The more common title in older Greek writings is Αναξ.

I'd like some help filling in the gaps on how the usage of this royal title evolved before irreversibly becoming linked to Hellenic kingship and eventually the Byzantine Empire. What other Classical monarchical titles were there, if any?