When I am at the server details page, there is a hyperlink extension that says "claim server" next to the Server Manager spot. When I click on it, it brings me to Server Management, but I'm not really sure what to do now. I've tried a few things but it still say to claim the server.

After clicking that button, you need to scroll down a little bit to the Claim Ownership button. You can click the (?) button near it to learn how to claim a server.

There is no Claim Ownership button. Once I click on "claim server," it shows me server management options.

You shouldn't need to claim a server again if you've already claimed it.