Search: Customizer preview when switching post types

In #8552, I started work on improving the customizer preview. But, I wasn't able to figure out how to either:

- Reload the search results automatically when the user changes the post types for the widget
- Show a notice to the user that the post types had changed and the search would need to be run again

Start by reading issue #8552 and tracing the customizer preview and search widget entry points. Determine whether changing post types should reload the search results or show a notice requiring another search; done means the selected behavior works when post types change.

Written by the indexing model from the issue text.