feat(ui): Add @me filter button, DM filter push-down, and DM FAB toggle

- Add person icon button in filter bar that inserts the current device
  name into the search field, for filtering own messages
- DM filter bar already benefits from the CSS sibling push-down rule
  added in previous commit (same class names used)
- Add collapsible FAB toggle to DM view, same pattern as channel chat

Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>

app/static/css/style.css

@@ -1311,6 +1311,15 @@ main {
     transition: background-color 0.15s ease;
 }
 
+.filter-bar-btn-me {
+    background-color: #e7f1ff;
+    color: #0d6efd;
+}
+
+.filter-bar-btn-me:hover {
+    background-color: #cfe2ff;
+}
+
 .filter-bar-btn-clear {
     background-color: #f8f9fa;
     color: #6c757d;