feat: rework board switcher (#6660)

* feat: rework board switcher

* fix: address board switcher review

apps/docs/docs/getting-started/after-the-installation.mdx

@@ -132,8 +132,8 @@ Here you have the following options in the header:
 - **Search**: Here you can search for data. You can search for integrations, boards, actions, sites and much more. Some actions may be context dependent (eg. will only show when certain widgets are on your page)
 - **Edit board**: Clicking this button will toggle edit mode. In this mode you can edit the contents of your board.
 - **Settings**: Clicking this button will redirect you to a settings page where you can adjust the board settings (page title, colors, ...).
-- **Board switcher**: Use this button to open another board that you can access.
-- **Profile**: This avatar will be visible on every page. Clicking it will show a menu with actions.
+- **Profile**: This avatar will be visible on every page. Its menu includes the board switcher; press `Shift+C` to open
+  it directly. The current board is left out of the results.
 
 When entering the edit mode, you'll get some additional buttons to work with:
 
@@ -369,7 +369,7 @@ Each page highlights the list view and the create action so you know where to fi
 
 ### Board tour
 
-When you first open a board, a tour highlights the board header elements: search, edit mode, settings, board switcher, and user menu.
+When you first open a board, a tour highlights the board header elements: search, edit mode, settings, and the user menu.
 
 ### Tour controls