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It's not my area of expertise (caring about the royal family), but I believe it was a well known nickname of hers: "Princess Di".
Or perhaps it's just a case of Chinese Whispers, and "Di" was simply the last thing her wicked stepmother shouted at her before she left for France. The butler must have overheard this, and naturally figured it was a nickname.
I see your confusion though, it's not an obvious one if you aren't familiar with her.