In my humble opinion no realm is an empire, and no ruler an emperor, unless the realm is a Roman Empire and the ruler a Roman Emperor.
Before 800 AD, if someone claimed to be an emperor, and his realm was known as the (or a) Roman realm, he was claiming to be a Roman Emperor and thus MIGHT have been a Roman Emperor.
In 800 AD Charlemagne was proclaimed Emperor in western Europe. After war and negotiations he was recognized as a basileus by the Eastern Roman or "Byzantine" Emperor. The word Basileus originally meant a simple king but by then meant semi-emperor or even full emperor.
The eastern Emperor then changed his title in Greek from Basileus, meaning (really great and powerful and semi-imperial) "King" to Basileus kai Autokrator ton Rhomaion (really great and semi-imperial) "King and Emperor of the Romans".
The Emperors of the Holy Roman empire never called themselves Holy Roman Emperors, but instead usually called themselves Imperator Romanorum et semper Augustus "Emperor of the Romans and always Augustus".
So after about 800 Ad it was pretty clear who was claiming to be a Roman Emperor and thus might be one.
In Latin the word Imperium originally meant authority and power, and more specifically the military, political, and legal command and authority granted to some Roman magistrates, usually with geographical and time limitations. An Imperator was a generic term for a possessor of imperium, especially a magistrate with imperium.
The Early Roman Emperors had three claims to the title imperator.
1) Augustus, the first Emperor, was granted the name and/or title of Imperator by the Senate, which was used by later Emperors. Most of the early Emperors used names in the form of Imperator Caesar (insert full name) Augustus with Imperator, Caesar, and Augustus gradually turning from names into titles meaning "emperor".
2) Emperors continued the earlier practice of being called Imperator by their troops after battle and thus claiming the right to triumphs. Thus Emperors often listed the number of times they were acclaimed Imperator in their full titles.
3) Among the most important powers granted to emperors was the imperium maius, or imperium proconsulare maius et infinitum, the "greater and infinite proconsular imperium" which made them direct governors of all provinces except for those few reserved for the senate, and with superior authority to proconsuls appointed by the senate, thus making the emperor governor of everywhere.
Since the emperor had more Imperium than all other magistrates combined, it was logical to call him THE Imperator, meaning the greatest and most powerful Imperator of them all.
Thus the English word emperor, derived ultimately from imperator is a good word for a Roman Emperor.
Obviously by this stricter definition the Persian Empire was not an empire, because it didn't even claim to be a Roman Empire.
But by a boarder definition, the Persian Empire was an empire because it was a realm with a political ideology which said it was the rightful government of the whole world/universe. And the Persian empire at its largest is estimated to have ruled over 44 percent of the total world population, coming closer to uniting all of humanity than any other realm in history, thus briefly making it the greatest empire ever.
Of course the Persian monarch did not call himself an emperor. The title of the Persian monarch during the Achaemenid dynasty was sometimes "the king" for short, which may have meant THE KING implying the king of everywhere, and was usually given in full as "The great King, the King of Kings, the King of Lands and Peoples, the King of the World". In the later Sassanid Dynasty the title was "King of Kings of Iran and of Non-Iran".
Thus the Persian monarch used a long and complicated title to claim to be the rightful ruler of everyone and everywhere, but for the Roman Emperor Imperator, or Caesar, or Augustus, or a combination of them, came to imply the same thing with just one or a few words.
Note that throughout history the title of king of kings has usually been much lower than emperor. There were a number of kings of kings who were subordinate to Roman Emperors at various times, and in the 20th century the Indian Empire had several subordinate rulers using the title of maharajadhiraja or "great king of kings".
Thus the Persian monarchs were the only kings of kings who could be considered emperor equivalents or emperors for short.
So Darius III could be considered an emperor or emperor equivalent. But what about Alexander?
Alexander was the King of Macedon and the leader of the Hellenic League or League of Corinth which was formed to invade the Persian Empire. The excuse for the invasion was revenge for burning the temples of the gods on the Acropolis during the invasion under Xerxes I. But it was certainly just for Xerxes to burn the temples of the gods to punish them for permitting their worshipers to break their oaths and revolt against the Persian Empire. And thus there was no justification for revenge for that.
Alexander managed to conquer the Persian Empire and rule it for a few years. The realm broke up a few years after Alexander's death as his generals and officials fought to seize kingdoms for themselves out of it. Thus the final result of Alexander's wars was to dissolve the mighty Persian empire into a number of warring realms.
Alexander's life was in constant danger for years as he marched and fought for thousands of miles alongside those same traitorous generals and officials who would later destroy the unity of the realm. Alexander had plenty of time to worry about what would happen after his death, which could have come at any moment, and to recognize the evil characters of his subordinates.
So if Alexander cared anything about his duty to the subjects of the Persian Empire he hoped to become ruler of, he would have come up with a plan to prevent his evil followers from destroying the empire after his death, or else he should have refrained from invading and conquering the Persian Empire.
Certainly the reign of Darius III could hardly have been as fatal for the Persia Empire if Alexander had not invaded, as Alexander's invasion turned out to be.
So I do not consider Alexander the Great to have been an emperor, but instead I consider him to be an evil rebel anti-emperor.
