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/Keeping-Agents-Running.md

@@ -30,9 +30,12 @@ output silence, and token stability.
 **For the other CLIs** it depends on what the CLI tells Codeman. Codex, OpenCode, Gemini
 CLI, Pi and OMP each declare their own prompt marker and working line, so they get the same
 screen check Claude does (before 1.26.1 every Codex session reported idle for its whole
-life). DeepSeek Harness reports idle, working and blocked to Codeman itself, which is as
-precise as hooks. Antigravity and Grok Build declare neither, so they fall back to output
-stabilization: the session is idle when output stops changing, which is coarser.
+life). DeepSeek Harness reports its end of turn and its dialogs to Codeman itself, which is
+as precise as hooks; its working and idle display still comes from its output. Antigravity
+and Grok Build declare neither. Codeman calls a new pane of theirs idle 3 seconds after
+launch, and after that it has only generic cues to go on (Claude's `❯` prompt marker, a
+braille spinner glyph), which these CLIs may not draw, so their working and idle status is
+unreliable.
 
 ## The Respawn Controller
 
@@ -163,7 +166,8 @@ run summary and lifecycle log carry the detail.
   a vague next cycle.
 - **Non-Claude sessions can respawn**, but with no usage-limit auto-resume, and with
   output-based idle detection except on DeepSeek Harness, whose own end-of-turn report
-  reaches the respawn controller the way Claude's `stop` hook does.
+  reaches the respawn controller the way Claude's `stop` hook does while its status bridge
+  is on (a local session, by default).
 - **Do not run respawn on a session you are actively typing in.** It will send prompts
   underneath you.