The Achaemenid era did not really have a single name for the empire; official ideology from Dareios on speaks of the Great King's xšaça, dominion, extending over many (twenty-odd) dahya, lands. This functions as an (arguably more modest, elegant and terse) variant on traditional Mesopotamian claims along the lines of "King of All" or "King of all the many inhabitants of every far-flung land" (I don't have my books at the moment, I can check the phrasing tomorrow). Cyrus just used the same stylings as Nabonidus and Aššurbanipal.

For the Arsacid/Parthian era, I don't think any specific style is attested beyond royal stylings like "King of Kings", "Philhellene", etc.

It's in the Sasanian era we begin to see the emergence of the kingdom as Ēranshahr, the Dominion of the Aryans. Shahr is the MP equivalent of xšaça; while Dareios notes himself as an Ariya, of Ariya descent, the Achaemenids did not use Aryan further than that (it may even be that calling oneself an Arya was akin to claiming noble blood). The Sasanian idea is more distinctly religious, being shaped by Zoroastrian traditions of being successors to an ancient land ruled by prehistoric Iranian kings. Eranshahr was eventually shortened to Eran.

I don't know to what extent Iran was used officially in the pre-Mongol Islamic era if at all, but Ghazan Khan declared himself Padishah of Iran and Islam in the 13th century, and the Safavids and Qajars both used Iran as a proper name to refer to their realms, so it was never forgotten; it may be that it was confined to Middle Persian tradition during the early Caliphates in favour of terms like Ajam (literally Land of the Mumblers) in Arabic and New Persian. Someone more familiar with Arabic sources would have to confirm this for me.