The names are not really important. That’s more for our conviniance. Server name and document root we really care about. Nginx just loads the config files. You can have eveyrthing in one file if you wanted to.
So I would leave the default as is. If that’s your main website, then just copy that config file, name it
old.site.config if you want, then just add the code above, responding to appropriate domain. It’s really up to you how you want to structure it.