Anon already picks a random keypair and adds an anon tag. Private just adds an encrypted note in the anon tag which has the nice benefit that it is backwards compatible and just looks like an anon payment to clients that haven't implemented it yet.

The amount is still public, this is just a modification to the zap request inside the zap invoice which gets included in the zap.

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Not sending a zap at all has the best privacy, but definitely not as fun.

Actually now that I think about it, private zaps will actually be more private when adding a comment since WoS and other wallet providers won't be able to read it.