Original Faction names

[Ludens]

Egypt are the Ptolemaioi.

Alexander the Great conquered Egypt and it was a part of his empire. When he died his empire was torn up between squabbling generals. Ptolemy took Egypt, therefore it is called Ptolemaioi.

[Nitpicking-mode on] Egypt was still called Aegyptus in EB's time. Ptolemaioi refers to the dynasty, not their possessions. It also isn't entirely correct to say that Egypt became Greek: the Ptolemaioi were not very successful at importing Greek settles (at least, compared to the Seleucids) and the original Egyptians mostly clung to the old ways. In fact, the Ptolemaioi kings went partially native.

To a lesser extent, this also applies to the Seleucids and Bactrians. They had a Hellenic ruling class and their subjects were influenced by Greek culture, but the population never became Greek, despite the presence of large colonies of Greek migrants. Nor was the cultural exchange entirely one-sided.

I am not certain the Greeks of Alexander's own time would approve of your description of him as Greek, to them he was but a barbarian with a sheen of Hellenism as far as I know.

It depends upon who you would ask. The Athenian orator Demosthenes considered the Macedonians and their king as semi-barbarians. However, one of his political rivals called the Macedonians "fellow Greeks". Whether or not the Macedonians were proper Greeks, their culture was certainly Greek (or Hellenistic, if you want to be exact). Once the Macedonians generals were in charge of most of the civilized world, the Macedonians-not-Greeks argument muted somewhat: if you can't beat them, join them, the clever Greeks must have thought. Still, there was quite a bit of rivalry between the "proper" Greeks, as the original inventors of the culture, and those that were responsible for expanding it to include most of the civilized world. In EB, this is reflected by Macedonian faction members becoming "mishellens" or "pro-hellenes", depending on their attitude to the "original" Greeks.