feat(live): polish — ended-state, chat, viewer actions, Private option

Restores the live streaming controllers / views / model from the
pre-audit WIP that got stashed on 2026-10-21, then applies the Phase 6
polish items:

## /live/new form
Add a Private visibility option that mirrors the upload modal's
wording — "Only you can see this stream. Perfect for testing without
an audience." Validator in LiveStreamController@store now permits
public|unlisted|private (was public|unlisted).

## Watch page ended-state
When the stream status is "ended", the overlay is a solid "Stream
ended" card showing the stream's duration (started_at → ended_at)
and, when a recording was published, a Watch Replay CTA linking to
the saved video. Previously an ended stream showed the generic
"Waiting for the broadcast to start" placeholder forever.

## Channel name link
The uploader's name in the meta row is now a real <a href> to the
channel page.

## Viewer action row
Subscribe + Share buttons below the player, visible to anyone except
the owner. Subscribe for guests uses the Phase 7 ?intended= pattern
so after sign-in they land back on the stream.

## Live chat (DB-backed polling)
New live_chat_messages table (migration 2026_10_21_000003). Viewers
GET /live/{stream}/chat?since=N every 3 seconds; signed-in users POST
to the same path (rate-limited at 60/min). Guests see the panel in
read-only mode with a "Sign in to chat" placeholder.

Private streams reject chat for anyone but the owner (404). Non-live
status rejects chat sends (422). Author name is snapshotted at post
time so a renamed account's old messages keep the name they used.

Co-Authored-By: Claude Opus 4.7 (1M context) <[email protected]>

app/Models/LiveChatMessage.php

@@ -0,0 +1,34 @@
+<?php
+
+namespace App\Models;
+
+use Illuminate\Database\Eloquent\Model;
+
+class LiveChatMessage extends Model
+{
+    protected $table = 'live_chat_messages';
+
+    public $timestamps = false; // explicit created_at only, set via useCurrent()
+
+    protected $fillable = [
+        'livestream_id',
+        'user_id',
+        'author_name',
+        'body',
+        'created_at',
+    ];
+
+    protected $casts = [
+        'created_at' => 'datetime',
+    ];
+
+    public function livestream()
+    {
+        return $this->belongsTo(LiveStream::class, 'livestream_id');
+    }
+
+    public function user()
+    {
+        return $this->belongsTo(User::class);
+    }
+}