SSH keys have two parts, the secret/private key (usually in
~/.ssh/id_rsa), and the public key (
~/.ssh/id_rsa.pub). The secret key can be used to prove who you are (or at least that you hold that secret key), and the public key can be used to check the secret key.
You never pass the secret key to any other party, as that would give them the ability to impersonate you.
As for which public key you install (or send to be installed) on the remote server, is up to you: it depends on what private key you want to use to login there.
If you have a private key on your Macbook, and want to login using that, then send the public key corresponding to that. That's probably the one in your first server's
authorized_keys or in your Macs
id_rsa.pub. If you want to login from the first server, then send the public key of that server's key, the one in the machines
If you want to login using both keys, you'll need to arrange both in the
authorized_keys on the target server.
If you wanted to, you could create multiple private keys on the same system and use different ones for different remote systems. That just requires a bit of bookkeeping to know which key you used where, and some configuration of the SSH client so that it knows to try to use all of the keys. (If you have lots of keys, you may need to configure it per-host.)
Passing the public key (
id_rsa.pub) to a third party is no risk. It's in fact exactly what you need to do to allow them to identify you by your private key.