Show the model and mode the session has, not the ones it was asked for

Picking either from the phone wrote the choice straight into the
session's state and then sent the request. Asking and having are
different things, and the difference is not rare: `auto` is a permission
mode the CLI accepts on the command line, silently resolves to
`default`, and refuses outright over the control channel -- "auto mode
unavailable for this model" -- so a session spawned in auto was in
default and one switched to auto stayed where it was, with the phone
reporting auto in both cases.

So the drivers report what they are set to and the manager follows that.
Measured, because the confirmations are not uniform: a model change
answers success with no value, so what was asked is remembered until the
answer arrives; a mode change echoes the mode it became, and that answer
wins over the request; and `init` names both -- resolving `haiku` to
claude-haiku-4-5-20251001 -- which also covers a session adopted from a
terminal that set them outside this app. A driver that cannot change
either already says so with an error, and now that error is the whole
story rather than a note beside a display that changed anyway.

The config keeps the requested value, deliberately: that answers a
different question, which is what to launch this session with next time.

Two things fall out. Control request ids are random rather than the
clock, because two in the same second shared an id and something now
looks them up. And the phone shortens a resolved name for the button --
`haiku-4-5` -- since the full one is what the CLI reports and roughly
twice the room that row has once Stop is in it.

app/androidApp/src/main/kotlin/com/example/aiapp/Events.kt

@@ -48,6 +48,14 @@ sealed class SessionEvent {
 
     data class Status(val state: String) : SessionEvent()
 
+    /**
+     * What the session is set to, as the session itself reports it.
+     *
+     * Either field alone: the two are confirmed separately and by different things. Asking for a
+     * change is not having one, so this -- not the request -- is what the pickers show.
+     */
+    data class Settings(val model: String?, val permissionMode: String?) : SessionEvent()
+
     data class UsageDelta(val tokens: Long) : SessionEvent()
 
     /**
@@ -108,6 +116,11 @@ fun parseSeqEvent(json: String): SeqEvent {
             "peerMessage" ->
                 SessionEvent.PeerMessage(body.getString("from"), body.getString("text"))
             "status" -> SessionEvent.Status(body.getString("state"))
+            "settings" ->
+                SessionEvent.Settings(
+                    model = body.optString("model").ifEmpty { null },
+                    permissionMode = body.optString("permissionMode").ifEmpty { null },
+                )
             "usageDelta" -> SessionEvent.UsageDelta(body.getLong("tokens"))
             "compacted" ->
                 SessionEvent.Compacted(