Nasu talks about the difference between them in HF BD2 interview
Alter servant = literally the same servant with entirely same saint graph, but an alternative version of them. Just a warped version of the hero.
Alter Ego = They are made from a different servant/god's saint graph, but they are their own existence, they have their own saint graph, they are themselves, they are not a copy of that servant. They are a new existence born from that hero. A class for bushins and other similar things. You are NOT the same person, but there's a connection to be made.
And then Takeuchi says "but that doesn't make sense with Okita Alter" and Nasu is "Whatever lol"
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