Pills only with the controls, web links in their final layout at once, chat pop-ups when paused by default

- Next episode (known ending, else the last two minutes) and Skip ED
  show only while the controls are up, like Skip 1:30 (PillGate).
- A web link's name and wide cover slot show from the first state, so
  nothing flips when its thumbnail arrives.
- Chat pop-ups default to Only when paused.

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>

apps/desktop/lib/src/screens/session/jellyfin_widgets.dart

@@ -36,9 +36,8 @@ class EpisodeSkipButton extends StatelessWidget {
   }
 }
 
-/// Floating "Next episode ›" from the ED (or the last two minutes):
-/// faint at rest, lit on hover. Lives outside the auto-hiding controls, so
-/// it shows in fullscreen and with the controls hidden.
+/// Floating "Next episode ›" from the ED (or the last two minutes), while
+/// the controls are up (`PillGate`): faint at rest, lit on hover.
 class NextEpisodePill extends StatelessWidget {
   final EpisodeLink episode;
   final VoidCallback onPressed;

apps/desktop/lib/src/screens/session/player_controls.dart

@@ -526,8 +526,8 @@ class SkipForwardButton extends StatelessWidget {
 }
 
 /// Floating pill over the video (Next episode, Skip OP / ED / 1:30):
-/// faint at rest, lit on hover. Lives outside the auto-hiding controls, so
-/// it shows in fullscreen and with the controls hidden.
+/// faint at rest, lit on hover. Lives outside the auto-hiding controls
+/// (also in fullscreen); `PillGate` shows it with them.
 class FloatingPill extends StatefulWidget {
   final String label;
   final String tooltip;