Public login means that you will use the Public login server hosted by EqEmulator.net. This will make it so anyone with an account registered through EqEmulator.net can log into your server. The Public Login server is a secure server requiring a username and password for authentication.
Minilogin means that you will run your own login server named Minilogin.exe. This is used primarily for Lan connections and authenticates based on IP address. This means that the server tells what user you are based on your IP address and not the username and password.
I hope this helps explain that a little better.
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