fix(tasks): correct the change-feed contract and its documentation

Review found the `onTasksChanged` teardown reasoning inverted. The comment
claimed every registration is an effect on the registry's own fiber, so
listeners would be gone before service disposal empties the store — but the
traceable proxy rebinds `this.ctx` to the CALLER, which this package's own
HMR-safety test already proves. The only shipped consumer registers from the
api-proxy mux stream, so it was still listening and simply kept the rows it
last received. Service disposal now announces the emptied set, and teardown
announces its stopping transition immediately instead of leaving an observer
on `running` for however long a slow producer takes to release.

Two documentation claims were false in the opposite direction: the Agent Note
and the ui-task README both said an unowned task is invisible in the header,
while `list(caller)` returns unowned tasks to every caller, the carrier fans
their changes out to every subscribed session, and this PR's own test asserts
exactly that. The note even contradicted itself two sections earlier. Both
sides now state the real asymmetries — another session's tasks, and the
process-local registry emptying on restart.

The "no Web path calls the consuming `ctx.tasks.read()`" invariant claimed a
test that did not exist; the carrier suite's producer had no `readOutput` at
all, so a stray read would have failed nothing. Its producer now counts cursor
consumption and the lifecycle and baseline paths both assert zero.

Also: a session created after the mux opened now receives the task baseline it
missed, the popover samples its clock when it opens rather than at mount, and
a failed task's unbounded producer detail elides instead of widening the row.

.agents/notes/implemented/feature/2026-08-08-web-background-task-display.md

@@ -125,7 +125,7 @@ Below it, [`tasks-local`](../../../../packages/tasks/tasks-local/tests/tasks.spe
 
 **Unowned-task fan-out is easy to under-implement.** Pushing only to the changed owner's session is correct for owned tasks and silently wrong for unowned ones, which are visible everywhere. The bug would surface only in compositions that create unowned tasks, which is why the carrier suite covers it directly.
 
-**The UI set is not the registry's set.** An unowned task is invisible in the header while `task_list` still reports it to the model. In practice every tool call carries an agent, so this stays theoretical, but the two surfaces are not interchangeable.
+**The UI set is not the registry's set.** The header shows what one session can see, so a task owned by another session never appears in it even though the registry holds it — and because the registry is process-local, a restart empties every list while the transcript still shows the `run_in_background` cards that started them. Unowned tasks are the opposite case: they reach every session's list, exactly as `list(caller)` reports them to every caller.
 
 **Settled rows accumulate.** The registry retains settled tasks until owner disposal, so a long session with many background commands grows a long list. Capping the settled tail is a presentation change, not a protocol one, if it becomes a real complaint.