Rename code mode tool to exec (#14254)

Summary
- update the code-mode handler, runner, instructions, and error text to
refer to the `exec` tool name everywhere that used to say `code_mode`
- ensure generated documentation strings and tool specs describe `exec`
and rely on the shared `PUBLIC_TOOL_NAME`
- refresh the suite tests so they invoke `exec` instead of the old name

Testing
- Not run (not requested)

codex-rs/core/src/codex.rs

@@ -409,7 +409,7 @@ impl Codex {
             && let Err(err) = resolve_compatible_node(config.js_repl_node_path.as_deref()).await
         {
             let message = format!(
-                "Disabled `code_mode` for this session because the configured Node runtime is unavailable or incompatible. {err}"
+                "Disabled `exec` for this session because the configured Node runtime is unavailable or incompatible. {err}"
             );
             warn!("{message}");
             let _ = config.features.disable(Feature::CodeMode);