So I would suggest the following:
Corresponding with the administration of various protected areas, I will notify them, that they should mark all official ways as what they are and provide visitors with the information about trespassing rules. This can then be mapped. Additionally, they should include following tags within relations on official ways:
access=yes
informal=no
foot=yes (for only hiking trails)
foot=designated and bicycle=designated (for hiking and cycling trails)
bicycle=yes (for cycling trails)
additionally they should tag ways that shouldn’t be used with informal=yes
The next step would be to discuss that with the companies providing app-based trail planning to see, if there are solutions or not.