docs: document all available search filter options

fern/products/docs/pages/customization/search.mdx

@@ -1,31 +1,41 @@
 ---
 title: Search configuration
-description: Configure search for your Fern docs using Algolia DocSearch. Filter by versions, endpoints, guides, and changelogs for better user experience.
+description: Configure search for your Fern docs using Algolia DocSearch. Filter by product, version, content type, API type, HTTP method, status code, and availability.
 ---
 
 <Markdown src="/snippets/agent-directive.mdx"/>
 
 Fern uses [Algolia DocSearch](https://docsearch.algolia.com/) to power search for your documentation. 
-DocSearch is designed specifically for documentation sites to help users quickly find what they need.
+DocSearch is designed specifically for documentation sites to help users find what they need.
 
 ## How search works
 
 DocSearch scans your Fern site's content and builds an index to generate search results. 
-It includes built-in filters that let users refine their searches by content type:
-
-- **Versions:** For sites with separate documentation for different API versions
-- **Endpoints:** Filters results by API reference documentation
-- **Guides:** Filters results by non-API reference documentation
-- **Changelog:** Filters results by changelog updates
+It includes built-in dropdown filters that appear dynamically based on your site's configuration, letting users refine their searches:
+
+- **Product:** Narrows results to a specific product in your documentation (for sites with multiple [products](/learn/docs/configuration/products))
+- **Version:** Filters results by documentation version (for sites using [versioned docs](/learn/docs/configuration/versions))
+- **Content type:** Filters results by guides, changelog entries, or API endpoints
+- **API type:** Filters API results by protocol (HTTP, webhooks, WebSockets, or gRPC)
+- **HTTP method:** Filters API results by HTTP method (`GET`, `POST`, `PUT`, `DELETE`, etc.)
+- **Status code:** Filters API results by HTTP status code
+- **Availability:** Filters API results by availability status, including Stable, Beta, and Deprecated
+
+<Frame caption="Fern's own docs site shows Product, Content type, and HTTP method filters. Other filters, like Version, don't appear because this site doesn't use versioned docs.">
+  <video
+    src="search.mp4"
+    autoPlay
+    loop
+    playsInline
+    muted
+  >
+  </video>
+</Frame>
 
 If you're using [Ask Fern](/learn/docs/ai-features/ask-fern/overview) (AI search), the search box also functions as your site's chat window.
 
-<Frame>
-<img src="./filtering.png" />
-</Frame>
-
 <Note>
-    **Note:** If an article includes the `nofollow` or `noindex` [frontmatter](/learn/docs/configuration/page-level-settings#indexing-properties), it will not be indexed by Algolia DocSearch and won't appear in search results.
+    **Note:** If an article includes the `nofollow` or `noindex` [frontmatter](/learn/docs/configuration/page-level-settings#indexing-properties), it won't be indexed by Algolia DocSearch and won't appear in search results.
 </Note>
 
 ## Integrating with Algolia