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FYI from the URL you gave you published it to Tableau Public, not Tableau Online. Tableau Public is free, Tableau Online is a SAAS version of Tableau Server.
Here are the instructions for publishing to Tableau Server or Tableau Online using Tableau Desktop, you'll need the URL for your Server or Online instance (and potentially login info if not using Active Directory) to proceed: Publish a Workbook.. You can also publish workbooks via Tableau's REST API and from the command line using the tabcmd tool.
It's not entirely clear what you're asking here. When you say "publish to an internal server", what exactly do you mean? Do just want to make the visualization available on a web server or something like that? If so, then you can easily embed it using embed code. If something else, then we may need a bit more clarification.