Cover the real provider switch and share chat Settings and stop checks

- Host test switches AI Provider from chat Settings through the real
  provider commands and checks the new mount keeps chat's context, frame,
  Back to chat and focus on the provider choice.
- Drop the open and minimized branches of the carried presentation: the
  provider choice lives only in Settings, so chat is never open then.
- One showChatSettings helper for the gear and the restore, and one
  isStoppable helper for the live-or-error check in chat and the panel close.

docs/CURRENT-STATE.md

@@ -767,8 +767,8 @@ starts. Settings opened from chat keep chat's frame and show the shared AI
 provider, the quick-commands option and then the voice preferences (Voice
 Preview, inactivity timeout and model tier), so they stay in reach while
 Preview is off, with Back to chat; voice Settings show Back to voice. Switching
-AI Provider keeps that view: chat's Settings, or chat itself, stay as they were
-for the new provider. Escape steps back from either Settings view, closes Voice
+AI Provider keeps that view: Settings opened from chat stay chat's Settings,
+in chat's frame, for the new provider. Escape steps back from either Settings view, closes Voice
 Preview, and does nothing in chat. Closing chat ends a typed session, and the
 voice panel's close turns voice off. Opening chat or closing the voice panel
 also clears a voice error left on the dock. Asking the assistant from chat to open