Remove ApiPrompt (#11265)

Keep things simple and build a full Responses API request request right
in the model client

codex-rs/codex-api/README.md

@@ -2,7 +2,7 @@
 
 Typed clients for Codex/OpenAI APIs built on top of the generic transport in `codex-client`.
 
-- Hosts the request/response models and prompt helpers for Responses and Compact APIs.
+- Hosts the request/response models and request builders for Responses and Compact APIs.
 - Owns provider configuration (base URLs, headers, query params), auth header injection, retry tuning, and stream idle settings.
 - Parses SSE streams into `ResponseEvent`/`ResponseStream`, including rate-limit snapshots and API-specific error mapping.
 - Serves as the wire-level layer consumed by `codex-core`; higher layers handle auth refresh and business logic.
@@ -11,14 +11,10 @@ Typed clients for Codex/OpenAI APIs built on top of the generic transport in `co
 
 The public interface of this crate is intentionally small and uniform:
 
-- **Prompted endpoints (Responses)**
-  - Input: a single `Prompt` plus endpoint-specific options.
-    - `Prompt` (re-exported as `codex_api::Prompt`) carries:
-      - `instructions: String` – the fully-resolved system prompt for this turn.
-      - `input: Vec<ResponseItem>` – conversation history and user/tool messages.
-      - `tools: Vec<serde_json::Value>` – JSON tools compatible with the target API.
-      - `parallel_tool_calls: bool`.
-      - `output_schema: Option<Value>` – used to build `text.format` when present.
+- **Responses endpoint**
+  - Input:
+    - `ResponsesApiRequest` for the request body (`model`, `instructions`, `input`, `tools`, `parallel_tool_calls`, reasoning/text controls).
+    - `ResponsesOptions` for transport/header concerns (`conversation_id`, `session_source`, `extra_headers`, `compression`, `turn_state`).
   - Output: a `ResponseStream` of `ResponseEvent` (both re-exported from `common`).
 
 - **Compaction endpoint**