Are the following asymmetric encryption schemes equivalent?
Generally, you should try to keep to standard practices unless the situation prevents you to do otherwise. That would mean that sign-then-encrypt should probably be used using two different key pairs.

Public keys are designed to be public and you should keep that in mind. Even if your scheme is secure, the system may not be. Public key operations could for instance leak the public key through side channel attacks. There is no need to protect the key after all. Key stores may not be available to store the key securely or at the right access level. Maybe you want to replace the software implementation with a hardware module that securely stores the private key - public keys won't be protected by such hardware.

An advantage of sign-then-encrypt is that you could decrypt the backup, then check the correctness of the signature afterwards. This allows you to store the backup decrypted somewhere without losing integrity. It also allows you to re-encrypt the package using a different key / scheme if that's required. Or to sign using a more fine-grained structure where parts of the backup are protected separately. In other words, using a separate signing key will give you a lot more flexibility.