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An English analogy might be with something like Tom or Katie. Sometimes, a person's birth certificate might even say Tom or Katie, but their Christian name would be Thomas or Katherine.
This may not seem like a big deal to us, as English speakers, but in Russia, this is very strictly followed. Alexy and Alexei are different names and all official Church sources would refer to him as Alexy. In deference to Russian sensabilities, I propose changing the article name to Alexy II, with a redirect to it from Alexei II, to deal with the common English misspelling.