docs: tighten the inbox, push and idle-detection wording

Follow-up to the previous commit, after a fact check against the code.

- Answer surfaces: the header bell answers Claude prompts, the phone
  strip only Claude dialogs (and only with the inbox on), and push
  Approve/Deny only a Claude permission prompt (PUSH_EVENT_MAP gives
  actions to permission_prompt alone). README, its Chinese mirror, the
  notifications page and the phone guide now say so.
- Antigravity and Grok Build: after the 3 s launch settle, idle is armed
  only by Claude's prompt marker and working only by a braille spinner,
  so "idle when output stops changing" was wrong. The wiki, the grok
  guide and the API reference now call their status unreliable.
- DeepSeek Harness: its idle and working reports become stop and
  agent_working, which drive respawn, waits and the inbox but not the
  session's own working/idle display. Keeping Agents Running says that,
  and that the respawn path needs the status bridge.
- The wait section of the API reference no longer calls exit the only
  dependable signal for screen-checked modes; the inbox page names how
  Codex and DeepSeek items clear; Core Concepts and Hooks And
  Integrations no longer call DeepSeek "the one" exception; Agent CLIs
  notes that Codex dialogs still raise the red alert; the inbox design
  doc notes the alert-only items.
- Skill: stop and blocked list deepseek, and SKILL.md no longer says
  codex has neither a transcript nor blocked. Mirrored with
  scripts/sync-plugin.mjs.

Co-Authored-By: Claude Opus 5.5 (1M context) <noreply@anthropic.com>

docs/wiki/Agent-CLIs.md

@@ -69,7 +69,7 @@ output. The other CLIs expose no equivalent.
 | Respawn cycling and unattended runs               | Yes    | Yes                                                  |
 | Cron jobs                                         | Yes    | Yes                                                  |
 | Docker cases, remote SSH cases                    | Yes    | Yes                                                  |
-| Precise idle detection                            | Yes    | Codex, Pi, OpenCode, OMP and Gemini: same screen check, via their own prompt and working line. DeepSeek: reports its state itself. Others: output stabilization, coarser |
+| Precise idle detection                            | Yes    | Codex, Pi, OpenCode, OMP and Gemini: same screen check, via their own prompt and working line. DeepSeek: reports its end of turn itself. Antigravity and Grok: generic cues only, unreliable |
 | Auto-resume when a usage limit resets             | Yes    | No                                                   |
 | Plan usage chip                                   | Yes    | No                                                   |
 | Approvals Inbox                                   | Yes    | DeepSeek and Codex as alerts (answer in the terminal); others no |
@@ -305,7 +305,8 @@ into the case's `.claude/settings.local.json` so that `/model` keeps working.
   auto-resume, the Approvals Inbox, and subagent visualization all assume it.
 - **Codex, OpenCode, Gemini, Antigravity, Grok, OMP** when you prefer that agent or that
   model. You get the session layer, respawn, cron, Docker, and remote SSH; you do not get the
-  hook-driven features.
+  hook-driven features (Codex dialogs still raise the red alert and an alert-only inbox card,
+  see [Codex](#codex) above).
 - **DeepSeek Harness** if you want DeepSeek's models with real status signals. It is the one
   non-Claude mode that reports idle, working and blocked to Codeman itself.
 - **Pi** if you want a fast, unsandboxed agent and you understand what project trust does.