Perhaps this is a ridiculous question, but how to I exit the login screen? After ssh-ing into my machine it asks for a login/password. It also boots me back there after logging in. How do I escape from this screen after logging off?

Sounds like you're connecting to LISH, the emergency console. Lish puts you in a screen session, so ctrl-A then "d" will detach it and put you back at the lish prompt. From there, you can "exit" (or "help", or "cake").

For day-to-day stuff, you probably want to ssh to your linode directly. That will do the right thing when you log out

So do I just want to do ssh <username>@<server>.linode.com, rather than
ssh linode<num>@<server>.linode.com ?

Oh wait, I got it. I want to ssh to the public ip. I'm dumb.

Don't fret... at least you now know what to do if you need to log in via the console.