fix(session): stop training the model to parrot [Message Time: ...]

Past assistant turns in history were prefixed with "[Message Time: ...]"
just like user turns. The model treated these as in-context demos and
started prefixing its own replies with the same marker, leaking
metadata to the user. Prompt-level warnings could not beat dozens of
prior assistant samples.

Annotate only user turns and proactive deliveries
(_channel_delivery=True, i.e. cron / heartbeat pushes whose timing is
the whole point and which are too infrequent to act as demos). Adjacent
user-side timestamps still pin every normal assistant reply for
relative-time reasoning. The now-redundant identity.md warning is
removed along with the demonstration source.

tests/agent/test_loop_save_turn.py

@@ -537,8 +537,12 @@ async def test_system_subagent_followup_is_persisted_before_prompt_assembly(tmp_
     non_system = [m for m in seen["initial_messages"] if m.get("role") != "system"]
     assert "question" in non_system[0]["content"]
     assert "working" in non_system[1]["content"]
+    # User turns carry the timestamp prefix so the model can reason about
+    # relative time. Assistant turns do NOT, otherwise the model treats those
+    # past replies as in-context examples and starts its own outputs with
+    # ``[Message Time: ...]`` (which then leaks back to the user).
     assert "[Message Time:" in non_system[0]["content"]
-    assert "[Message Time:" in non_system[1]["content"]
+    assert "[Message Time:" not in non_system[1]["content"]
     assert non_system[2]["content"].count("subagent result") == 1
     assert "Current Time:" in non_system[2]["content"]