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/deepseek-integration.md

@@ -265,9 +265,8 @@ siblings; no other external CLI mode qualifies. Two edges are worth repeating he
   reports a turn that never ran, and leaves the prompt in a pane that was not yet
   accepting input. Wait for the composer (`❯`) instead.
 - **Wait on `stop`, not on the default signal set.** That set also carries `idle`,
-  which for dsh, as for every external CLI that declares no working line, is
-  inferred from output stabilization; a dsh TUI that repaints rarely reads as idle
-  mid-turn.
+  which for dsh is inferred from output stabilization; a dsh TUI that repaints
+  rarely reads as idle mid-turn.
 
 ## 6. The web UI as a tab