In my understanding, to ssh login to my server, I need a private key:
However, the key I get when generating seems to be a public key:
Do you know where to locate my private key?
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In the line
Enter file... you have chosen an arbitrary filename
public_key_ooker. That's why the privake key has been saved to
public_key_ooker and public key to
public_key_ooker.pub. The name alone doesn't make the private key a public key, but gets you confused.
But you don't necessarily need a key, if you are using password authentication. If you'd rather use public key authentication, which is actually a good idea, the server needs to be aware of your public key: you need to save the line to
~/.ssh/authorized_keys on the server.