The origin of the belief that Mark went to Egypt

Did the belief that Mark was in Egypt derive from the assumption that 1 Peter 5:13 refers to the Babylon near Cairo?

1 Peter 5:13

She who is in Babylon, chosen together (συνεκλεκτὴ), sends you greetings; and so does my son Mark.

Cassiodorus's translation of Clement of Alexandria.

The second Epistle of John, which is written to virgins, is very simple: it is written indeed to a certain Babylonian lady, Electa [i.e., Eclecte] by name; but it signifies the election of the holy Church.

Clement understood (correctly — see Lincoln Blumell's recent book) that 2 John was written to a woman called Eclecte. Clement, or a predecessor, inferred that Eclecte was part of a community of virgins, perhaps because they balked at the idea that Eclecte had authority over men. Also, a Coptic manuscript, for example, says that 2 John was written to virgins. Clement found Eclecte also in 1 Peter 5:13 by reading σὺν Ἐκλέκτῃ (with Eclecte) instead of συνεκλεκτὴ, or by inferring that συνεκλεκτὴ alludes to the name of “she who is in Babylon”. Clement does not seem to identify Babylon as Rome, but there was a Babylon just 220 km (2.5 days by boat) from Alexandria. It is near modern Cairo and is known as Babylon Fortress. To someone in Alexandria “Babylon” might mean the Babylon in Egypt. The idea that Mark was in Egypt would perhaps appeal to Clement and others in Alexandria.

 

If Clement or a predecessor had this Babylon in mind, he thought that Mark was in Egypt during the lifetime of Peter, and that a community of virgins was there. Is this the origin of Eusebius's traditions that Mark was in Egypt before 62 and that there was a community of Christian virgins there?

 

Eusebius E.H. 2

16.1. And they say that this Mark was the first that was sent to Egypt, and that he proclaimed the Gospel which he had written, and first established churches in Alexandria

16.2. And the multitude of believers, both men and women, that were collected there at the very outset, and lived lives of the most philosophical and excessive asceticism, was so great, that Philo thought it worthwhile to describe their pursuits, their meetings, their entertainments, and their whole manner of life.”

17.3 In the work to which he gave the title, On a Contemplative Life or on Suppliants, after affirming in the first place that he will add to those things which he is about to relate nothing contrary to truth or of his own invention, he says that these men were called Therapeutæ and the women that were with them Therapeutrides

24 When Nero was in the eighth year of his reign, Annianus succeeded Mark the evangelist in the administration of the parish of Alexandria

Philo, On the Contemplative Life or Suppliants 68 describes the Therapeutrides:

And the women also share in this feast, the greater part of whom, though old, are virgins in respect of their purity. 

Babylon in Egypt

Strabo Geography 17. 1st Century BC

And, having sailed farther up the river, one comes to Babylon, a stronghold, where some Babylonians had withdrawn in revolt and then successfully negotiated for permission from the kings to build a settlement

Diodorus Siculus 1.56.3 1^st Century BC.

And it is said that the captives brought from Babylonia revolted from the king, being unable to endure the hardships entailed by his works; and they, seizing a strong position on the banks of the river, maintained a warfare against the Egyptians and ravaged the neighbouring territory, but finally, on being granted an amnesty, they established a colony on the spot, which they also named Babylon after their native land. 

John, Bishop of Nikiu. The Chronicle C. 690 A.D. 

And Trajan came to Egypt and built a fortress with a strong impregnable tower, and he brought water into it in abundance and he named it Babylon in Egypt. 17. Nebuchadnezzar the king of the Magi and Persians was the first to build its foundations and to name it the fortress of Babylon.