v5.2.0: Migrate to DocSearch 3 (#36176)

* Migrate to DocSearch v3

Fixes #33338

* initialize `siteDocsVersion` const after the proper check

* Update site/layouts/partials/stylesheet.html

Co-authored-by: GeoSot <geo.sotis@gmail.com>

* Update header.html

* code review changes

* Update search.js

* Fix contrast issue, tweak some styles

* Fix some navbar changes

* temporary safari fix for docsearch3

Co-authored-by: GeoSot <geo.sotis@gmail.com>
Co-authored-by: XhmikosR <xhmikosr@gmail.com>

site/layouts/_default/docs.html

@@ -62,4 +62,5 @@
   {{ range .Page.Params.extra_js -}}
     <script{{ with .async }} async{{ end }} src="{{ .src }}"></script>
   {{- end -}}
+  <div class="position-fixed"><input type="text"></div>
 {{ end }}