Hi, so in OSM the correct way to handle this is to use OSM's tagging system. access=no would be appropriate if it's officially closed to the public. Once this is done, services that do use OSM data will reflect that and specify their access. It might take some time to reflect in other apps.

I went ahead and changed the access tag to access=no for this trail. But of course if you haven't already, the best deterrent is a physical barrier and signs posted. Most people dont see a trail and check the map, they just walk that way if there is nothing stopping them. Good luck, when there is really no trace of the trail left, it can then be removed.