fix: isolate New Chat from in-progress web sessions

[sam-saffron-jarvis]

What

Fix New Chat state leakage from in-progress web sessions.

• Treat a send as an interjection only when the currently active, non-draft session owns the busy state.
• Keep inactive session active_run status in the sidebar without flipping the global draft composer into streaming mode.
• Add regression coverage for:
  • New Chat draft sends with stale global streaming=true
  • late active-run sync responses after switching to New Chat

Verification

mise exec node@24 -- node internal/serveui/static/app_stream_test.js
mise exec node@24 -- node internal/serveui/static/app_sessions_test.js
go test ./...
go build ./...