Move example scripts to the docs website

And make sure the level of testing doesn't drop (the examples are run as part of the test).

No files or test entry points are named in the issue. Locate the example scripts, the documentation website, and the existing test path that runs the examples; then determine how the move can preserve that coverage. Done means the examples are available on the docs website and their current testing level has not dropped.

Written by the indexing model from the issue text.