fix(extension): classify the home tab by id, not a racy boot flag (issue #57)

SessionManager.classifyNewTab's `initializing` branch relied on a
windowInitializing flag that start() cleared *before* the home tab's
chrome.tabs.onCreated event was dispatched (the event is asynchronous), so
the branch never ran in production and the home tab's classification was
subject to a timing race. Match the home tab by its tab id instead — the id
is already returned by ensureActiveTab and is reliable regardless of event
ordering.

Also refresh the overlay-bridge docstring: overlay.ready now makes the
background *push* the per-tab overlay state (and the Agent Window also
pushes on tab create / control-mode change), rather than a request/reply.

apps/extension/src/lib/overlay-bridge.ts

@@ -3,8 +3,10 @@
  * between the content-script control overlay and the background SW.
  *
  * Content script → background:
- *  - `{ kind: "overlay.ready" }` → background replies with the authoritative
- *      overlay state for the sender's window.
+ *  - `{ kind: "overlay.ready" }` → background decides the per-tab state first
+ *      (see overlayStateForTab) and pushes the authoritative overlay state for
+ *      that tab. It also proactively pushes on tab create and control-mode
+ *      change, so a tab usually receives state before it ever sends ready.
  *  - `{ kind: "overlay.interrupt", sessionId }` → background asks the
  *      daemon (via a `session.user_interrupt` WS event) to cancel
  *      every inflight + queued tool call for that session with