Sidebar widget not displaying after simple code update.

1. If you don’t need the others settings from register_sidebar, that’s fine.

2. That’s not fine. 🙂
   To render your sidebar you need to call

    dynamic_sidebar($index);
   

   In your case it would be:

    dynamic_sidebar('Right Sidebar');
   

   see the codex page for dynamic sidebar
   http://codex.wordpress.org/Function_Reference/dynamic_sidebar

   Your current code says (short version): If there is no sidebar, then do my custom code.
   Look at the examples from the linked codex page.

3. skipped

4. If you need those settings for a sidebar, use them, but notice, and that’s what happend on your sidebar/widget configuration page when your old sidebar disappeared.

If you register a sidebar with an ID, WordPress looks in the database for this ID, if you just register a sidebar with a name, WordPress will look for the name.

Your ID was different from the original name, and there was no entry for a sidebar with this id.
The name gets ignored, when an ID is set.

Hope that makes things a bit more clear.