This renames `$tax` to `$taxonomy_object` and `$s` to `$search` for clarity. The latter is only renamed when used as an internal variable and not referring to the `$s` global.
The list of affected functions:
* `wp_ajax_ajax_tag_search()`
* `wp_ajax_add_link_category`
* `wp_ajax_add_tag()`
* `wp_ajax_get_tagcloud()`
* `wp_ajax_inline_save_tax()`
* `wp_ajax_find_posts()`
Follow-up to [6542], [8901], [10222], [12833], [16771], [16992], [22723], [38698].
Props azouamauriac.
Fixes#55098.
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