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>

skills/codeman/SKILL.md

@@ -552,8 +552,9 @@ That is true of no other non-claude mode, and it is worth knowing why: the DeepS
 Harness TUI reports `idle`/`working`/`blocked` to Codeman over the supervisor contract it
 implements, so dsh is the one external CLI with definitive `stop`/`blocked` signals
 instead of guessed-from-silence ones — and it writes a structured transcript, which is
-what `last-response` reads for it. `shell`, `opencode`, `codex`, `gemini`, `antigravity`,
-`pi`, `grok` and `omp` have neither and still need markers ([§5.5](reference/verbs.md#55-markers-for-hook-less-workers)).
+what `last-response` reads for it. `shell`, `opencode`, `gemini`, `antigravity`, `pi`,
+`grok` and `omp` have neither, and `codex` has a transcript and a screen-read `blocked` but
+no `stop`, so all of them still need markers ([§5.5](reference/verbs.md#55-markers-for-hook-less-workers)).
 
 Three things to know before you spawn one:
 

skills/codeman/reference/endpoints.md

@@ -634,8 +634,8 @@ Three bounded long-polls. Shared semantics:
 |--------|---------|---------------|
 | `idle` | output stabilized + prompt detected, heuristic, can flap mid-turn | every mode |
 | `working` | session started producing output | every mode |
-| `stop` | Claude Code `stop` hook, the definitive end-of-turn | `claude` only |
-| `blocked` | `permission_prompt` / `elicitation_dialog` hook, the worker needs an answer | `claude`; also `codex`, whose approval, question and trust dialogs Codeman reads off its screen (answer them in the terminal, never through the inbox) |
+| `stop` | Claude Code `stop` hook, the definitive end-of-turn | `claude`; also `deepseek`, whose harness reports its own end of turn while its status bridge is on |
+| `blocked` | `permission_prompt` / `elicitation_dialog` hook, the worker needs an answer | `claude`; also `deepseek` (its harness reports its dialogs, same condition), and `codex`, whose approval, question and trust dialogs Codeman reads off its screen (answer both in the terminal, never through the inbox) |
 | `exit` | PTY exited or session deleted | every mode |
 
 ⚠️ **`claude` mode is necessary for `stop`/`blocked`, not sufficient. The real