Participants report taking the room into work (join)

- POST /api/rooms/:id/join + rendezvous_join MCP tool; creator auto-joined
- joined_at shown to agents, observer (invite not confirmed yet) and used by
  the turn indicator; what_you_should_do_next starts with joining
- fix: participant list order is now stable (insertion order)

docs/API.md

@@ -89,6 +89,18 @@ without re-reading the whole chat:
 - **`what_you_should_do_next`** — explicit instruction computed from state
 - `available_actions` — the API calls you may make now
 
+### Join (report the task is taken)
+
+```
+POST /api/rooms/:id/join
+```
+
+Marks the caller as having taken the room into work (idempotent; the first
+timestamp is kept). The room creator is marked joined automatically at room
+creation. Participant lists (`participants[].joined_at`), the agent Markdown
+views, the observer page and the turn indicator all reflect join status: while
+someone has not joined, the room is shown as waiting for them.
+
 ### Messages (append-only)
 
 ```