Compare Leo Tolstoy and Fyodor Dostoevsky. There are many possible axes of comparison, but I want to focus on their names. To his mother (and other compatriots), Tolstoy was and is Lev, since that's the Russian equivalent of Leo; yet to English speakers, he is Leo. Dostoevsky, on the other hand, could be Teddy. Well, OK, he wouldn't be, but he could be Theodore, since that's the analog of Fyodor; yet he remains Fyodor.
Likewise, compare Czar Nicholas (Nikolai) and Joseph Stalin (Yosif) with Mikhail — not Michael — Gorbachev. (I don't know why some people render Stalin's name as Josef, since it doesn't approach the Russian pronunciation any better; note also that while Stalin was born in Georgia, the name was borrowed into English from Russian, not from Georgian.) What's up there?
I have some suspicions, for instance relating to how similar the name is to the English version (which explains Lev becoming Leo, but Fyodor staying Fyodor, but doesn't explain Mikhail staying Mikhail), and relating to the time of English adoption. But I wonder whether others have studied this more closely, not just as to Russian but also as to other foreign-alphabet languages.
Related question: What about the occasional Anglicization of names from Latin-alphabet languages? The one example I know off the top of my head is Popes; is that just simply related to the Church's willingness to speak to people in their own language (even during the era when the Mass was in Latin), or is there more to it? Are there other examples?