This has been covered before in the Dedicated Server forum. It's always a good idea to do a Search before asking a question, as your question has probably been asked before.
When you make a server locally, it will always be in your Local list because.....it is local. You are connecting to it locally, not over the Internet. Pretty obvious, isn't it?
If you have succeeded in making your server visible to others over the Internet, it will be in their Internet section because.....they are conecting to your server over the Internet. Makes sense, doesn't it?
Again, please read or Search the other threads in the Dedicated Server forum. There is a lot of information there waiting to be discovered.