HTML search: adjustments to type-dependent CSS classnames and defaults (#12815)

Co-authored-by: Tim Hoffmann <2836374+timhoffm@users.noreply.github.com>

doc/development/html_themes/index.rst

@@ -238,16 +238,25 @@ Styling search result entries by category
 
 .. versionadded:: 8.0
 
+.. note::
+
+    The CSS classes named below are generated by Sphinx's standalone search
+    code.  If you are using a third-party search provider, such as
+    ReadTheDocs_, to provide search results, then the theming options available
+    may vary.
+
+.. _ReadTheDocs: https://docs.readthedocs.io/
+
 The search result items have classes indicating the context in which the
 search term was found. You can use the CSS selectors:
 
-- ``ul.search li.context-index``:
+- ``ul.search li.kind-index``:
   For results in an index, such as the glossary
-- ``ul.search li.context-object``:
+- ``ul.search li.kind-object``:
   For results in source code, like Python function definitions
-- ``ul.search li.context-title``:
+- ``ul.search li.kind-title``:
   For results found in section headings
-- ``ul.search li.context-text``:
+- ``ul.search li.kind-text``:
   For results found anywhere else in the documentation text
 
 As a base for inheritance by other themes, the ``basic`` theme is
@@ -265,16 +274,16 @@ search result list:
         padding: 5px 0 5px 10px;
         list-style-type: "\25A1";  /* Unicode: White Square */
     }
-    ul.search li.context-index {
+    ul.search li.kind-index {
         list-style-type: "\1F4D1";  /* Unicode: Bookmark Tabs */
     }
-    ul.search li.context-object {
+    ul.search li.kind-object {
         list-style-type: "\1F4E6";  /* Unicode: Package */
     }
-    ul.search li.context-title {
+    ul.search li.kind-title {
         list-style-type: "\1F4C4";  /* Unicode: Page Facing Up */
     }
-    ul.search li.context-text {
+    ul.search li.kind-text {
         list-style-type: "\1F4C4";  /* Unicode: Page Facing Up */
     }