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Saying SecretKey rather than PrivateKey is useful because it clears up the notation:

S_A(P_B(P_A(S_B(C)))) = C

is clearer than

Pri_A(Pub_B(Pub_A(Pri_B(C)))) = C


To the extent that that should be a deciding fact, I agree.

I guess it depends on how normative or descriptive one wants the term used to describe such a key.

Something private may remain private even if it is no longer secret. One might know some sad details of someone's marriage, for instance (ie, they are no longer secret between the parties) but those details still fall into the realm of the private. In other words, those details still merit some discretion in their handling because their private nature remains intact (I would say "personal" but that doesn't help us in discussing keypairs, since each key can be "personal").

The non-public key may not, indeed, be secret, and one should be ever mindful of the possibility that it is no longer so.

If one wants a normative label, then "secret" it is. If one wants a descriptive label, then "secret" might imply to the cryptographically naive more protection than is warranted (and if we aren't writing for the cryptographically naive, for whom are we writing these descriptions?)


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