Extract request_user_input normalization into codex-tools (#16503)

## Why
This is another incremental step in the `codex-core` -> `codex-tools`
migration called out in `AGENTS.md`: keep pure tool-definition and
wire-shaping logic out of `codex-core` so the core crate can stay
focused on runtime orchestration.

`request_user_input` already had its spec and mode-availability helpers
in `codex-tools` after #16471. The remaining argument validation and
normalization still lived in the core runtime handler, which left that
tool split across the two crates.

## What Changed
- Export `REQUEST_USER_INPUT_TOOL_NAME` and
`normalize_request_user_input_args()` from
`codex-rs/tools/src/request_user_input_tool.rs`.
- Use that `codex-tools` surface from `codex-rs/core/src/tools/spec.rs`
and `codex-rs/core/src/tools/handlers/request_user_input.rs`.
- Keep the core handler responsible for payload parsing, session
dispatch, cancellation handling, and response serialization.

This is intended to be a straight refactor with no behavior change.

## Verification
- `cargo test -p codex-tools`
- `cargo test -p codex-core request_user_input`

codex-rs/tools/src/request_user_input_tool.rs

@@ -3,8 +3,11 @@ use crate::ResponsesApiTool;
 use crate::ToolSpec;
 use codex_protocol::config_types::ModeKind;
 use codex_protocol::config_types::TUI_VISIBLE_COLLABORATION_MODES;
+use codex_protocol::request_user_input::RequestUserInputArgs;
 use std::collections::BTreeMap;
 
+pub const REQUEST_USER_INPUT_TOOL_NAME: &str = "request_user_input";
+
 pub fn create_request_user_input_tool(description: String) -> ToolSpec {
     let option_props = BTreeMap::from([
         (
@@ -78,7 +81,7 @@ pub fn create_request_user_input_tool(description: String) -> ToolSpec {
     let properties = BTreeMap::from([("questions".to_string(), questions_schema)]);
 
     ToolSpec::Function(ResponsesApiTool {
-        name: "request_user_input".to_string(),
+        name: REQUEST_USER_INPUT_TOOL_NAME.to_string(),
         description,
         strict: false,
         defer_loading: None,
@@ -105,6 +108,24 @@ pub fn request_user_input_unavailable_message(
     }
 }
 
+pub fn normalize_request_user_input_args(
+    mut args: RequestUserInputArgs,
+) -> Result<RequestUserInputArgs, String> {
+    let missing_options = args
+        .questions
+        .iter()
+        .any(|question| question.options.as_ref().is_none_or(Vec::is_empty));
+    if missing_options {
+        return Err("request_user_input requires non-empty options for every question".to_string());
+    }
+
+    for question in &mut args.questions {
+        question.is_other = true;
+    }
+
+    Ok(args)
+}
+
 pub fn request_user_input_tool_description(default_mode_request_user_input: bool) -> String {
     let allowed_modes = format_allowed_modes(default_mode_request_user_input);
     format!(