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/partials/stylesheet.html

@@ -1,4 +1,7 @@
-{{- "<!-- Bootstrap core CSS -->" | safeHTML }}
+{{ if eq .Page.Layout "docs" -}}
+<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/@docsearch/css@3">
+{{- end }}
+
 {{ if eq hugo.Environment "production" -}}
 {{ if eq .Page.Params.direction "rtl" -}}
 <link href="/docs/{{ .Site.Params.docs_version }}/dist/css/bootstrap.rtl.min.css" rel="stylesheet" {{ printf "integrity=%q" .Site.Params.cdn.css_rtl_hash | safeHTMLAttr }} crossorigin="anonymous">