Surface the session's first-class Project in both chat surfaces: the
Desktop status bar (project name as the workspace label, full cwd in the
tooltip) and the TUI status label + /status output.
One source of truth. The per-profile projects.db is the authority, read
in tui_gateway via _project_info_for_cwd (backed by
projects_db.project_for_path) and threaded through every session.info
emission path the TUI consumes. The Desktop already caches that truth in
$projectTree, so it DERIVES the label from it (projectNameForCwd) instead
of carrying a second per-session $currentProject atom fed from
session.info.
That drops the parallel state #64721 introduced and the entire
reset/reconciliation surface it required (resume, agentless cwd.set,
gateway-switch, fresh-draft): the label is purely derived, so it stays
correct whenever the cwd or the project tree changes. Only explicit,
named projects resolve on both surfaces, so an auto-discovered repo root
keeps the cwd-leaf label everywhere.
Excludes the unrelated markdown shell-fence change bundled in #64721.
