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My dictionary does not agree precisely, but I think you have the steps approximately correct.

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However, I'm not convinced that parents have this Old French spelling in mind when choosing the form Fae as opposed to Fay.

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You can check this page to see the usage in the US:
https://www.behindthename.com/name/fay/top/united-states/f?compare=fae%20faye&type=percent