Add public skills + improve repo skill discovery and error UX (#8098)

1. Adds SkillScope::Public end-to-end (core + protocol) and loads skills
from the public cache directory
2. Improves repo skill discovery by searching upward for the nearest
.codex/skills within a git repo
3. Deduplicates skills by name with deterministic ordering to avoid
duplicates across sources
4. Fixes garbled “Skill errors” overlay rendering by preventing pending
history lines from being injected during the modal
5. Updates the project docs “Skills” intro wording to avoid hardcoded
paths

codex-rs/docs/protocol_v1.md

@@ -68,7 +68,7 @@ For complete documentation of the `Op` and `EventMsg` variants, refer to [protoc
   - `Op::UserInput` – Any input from the user to kick off a `Task`
   - `Op::Interrupt` – Interrupts a running task
   - `Op::ExecApproval` – Approve or deny code execution
-  - `Op::ListSkills` – Request skills for one or more cwd values
+  - `Op::ListSkills` – Request skills for one or more cwd values (optionally `force_reload`)
 - `EventMsg`
   - `EventMsg::AgentMessage` – Messages from the `Model`
   - `EventMsg::ExecApprovalRequest` – Request approval from user to execute a command
@@ -77,6 +77,7 @@ For complete documentation of the `Op` and `EventMsg` variants, refer to [protoc
   - `EventMsg::Warning` – A non-fatal warning that the client should surface to the user
   - `EventMsg::TurnComplete` – Contains a `response_id` bookmark for last `response_id` executed by the task. This can be used to continue the task at a later point in time, perhaps with additional user input.
   - `EventMsg::ListSkillsResponse` – Response payload with per-cwd skill entries (`cwd`, `skills`, `errors`)
+  - `EventMsg::SkillsUpdateAvailable` – Notification that skills may have changed and clients may want to reload
 
 The `response_id` returned from each task matches the OpenAI `response_id` stored in the API's `/responses` endpoint. It can be stored and used in future `Sessions` to resume threads of work.