External session continued in both apps appears strange in both apps

Testing #331122

In this example, first message was sent in gh app, second was sent in vs code, third in gh app.

Messages sent from VS Code were not shown right away in the GH app, but when I closed and reopened GH I saw the message and it included some system-prompt-looking text. Unsure if this is even on us.

In this example the last message I sent from the GH app did not update in vs code until I reloaded the window.