Inference Providers: Add Responses API guide (#1980)

* Add Responses API guide

* Move provider pinning tip in Responses API guide

* Update Responses API guide for clarity and onboarding

* remove enterprise controls

even if it sounds cool

* "pin" -> "use"

* Fix string quoting in API example payload

* "Review" -> "Read"

* Add tip about model compatibility to Responses API docs

Author: gary149

docs/inference-providers/_toctree.yml

@@ -19,6 +19,8 @@
     title: Structured Outputs with LLMs
   - local: guides/function-calling
     title: Function Calling
+  - local: guides/responses-api
+    title: Responses API (beta)
   - local: guides/gpt-oss
     title: How to use OpenAI gpt-oss
   - local: guides/image-editor