_set_session_context() populated session id/source/cwd but omitted the
session's profile, so a non-launch Desktop session never exported
HERMES_SESSION_PROFILE to its tools; tools/terminal_tool.py reads it for
the docker container/profile slot.
Trimmed to the surviving Desktop-side change of #72480 — the local/PTY
terminal, docker, and execute_code surfaces landed on main via 7138b9587a.