[FEATURE]: Docs persist database type

### Describe what you want

In the docs, if you change the database type (i.e., PostgreSQL, MySQL, or SQLite) for one example, all other examples update to that database type.

Furthermore, the selection is persisted across multiple page views (perhaps in local storage).

Why? Because I'm using MySQL, I have to click on the MySQL tab for every single example and it's getting annoying.