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.

changelog.d/1546-chat-provider-switch.md

@@ -1,5 +1,5 @@
 - Keep typed chat's view when switching AI Provider: Settings opened from
-  chat, or chat itself, stay as they were for the new provider instead of
-  falling back to voice and widening the dock. Opening chat or closing the
+  chat stay chat's Settings for the new provider instead of falling back to
+  voice and widening the dock. Opening chat or closing the
   voice panel now also clears a voice error left on the dock (Manjunath,
   @manjunath22466, #1546).