The 700s show it clearly on the PIS screens, so can't the 377s and 387s be adjusted? There are certainly announcements and displays showing the coach number....there need to be screens on all units that clearly state whether first is in operation...
Usually I go by the ATOS announcements / PIS at the station. If the announcement or display says "1st class accommodation is not available on the train" or "1st class areas can be used by all ticket holders" then it's quite clear.