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/Mobile-Guide.md

@@ -43,9 +43,9 @@ as the keyboard appearing.
 
 Tapping the "C" logo gives a session overview rather than a welcome page:
 
-1. **NEEDS YOU** first: sessions blocked on a question, with answer strips so you can
-   resolve a Claude prompt without opening the session. A Codex or DeepSeek Harness row
-   offers **Answer in terminal** instead, which opens the session.
+1. **NEEDS YOU** first: sessions blocked on a question, with answer strips (Approvals Inbox
+   on) so you can resolve a Claude dialog without opening the session. A Codex or DeepSeek
+   Harness row offers **Answer in terminal** instead, which opens the session.
 2. **CURRENT SESSIONS** with live status.
 3. **PAST SESSIONS**, resumable.
 
@@ -169,8 +169,8 @@ phone, and it is the main reason the feature exists.
 ## Notifications
 
 Push notifications reach you with no tab open, and with the Approvals Inbox on, those for a
-Claude prompt carry **Approve** and **Deny** buttons handled by the service worker, so you
-can unblock a Claude agent from the lock screen.
+Claude permission prompt carry **Approve** and **Deny** buttons handled by the service
+worker, so you can unblock a Claude agent from the lock screen.
 
 Setup in [Notifications And Approvals](Notifications-And-Approvals).