The WP Site Search query variable is what the plugin uses to pass search terms to the search page. For most sites, the default q variable will work just fine but you can customize this variable if you'd like.
When the variable name is set to q (the default), and your search page has the slug /search (also the default), then searches will be performed using the following request URI:
/search/?q=[search terms]
When you configure WP Site Search so that it handles all WordPress searches automatically, then any time WP Site Search sees a request like: /?s=[search terms], that will automatically redirect a visitor to: /search/?q=[search terms]
If you set the variable name to query instead of q, then searches will be performed using the following request URI:
/search/?query=[search terms]
Note: Please do not use s here; it's reserved for WordPress itself.
