No, I’m asking for clarification, with a plausible interpretation.
As you found, I also found this provision of the Virginia criminal statutes (Va. Code, § 18.2-119. Trespass after having been forbidden to do so; penalties.):
If any person without authority of law goes upon or remains upon the lands, buildings or premises of another, or any portion or area thereof, after having been forbidden to do so, either orally or in writing, by the owner, lessee, custodian, or the agent of any such person, or other person lawfully in charge thereof, or after having been forbidden to do so by a sign or signs posted by or at the direction of such persons or the agent of any such person or by the holder of any easement or other right-of-way authorized by the instrument creating such interest to post such signs on such lands, structures, premises or portion or area thereof at a place or places where it or they may be reasonably seen, or if any person, whether he is the owner, tenant or otherwise entitled to the use of such land, building or premises, goes upon, or remains upon such land, building or premises after having been prohibited from doing so by a court of competent jurisdiction by an order issued …, and after having been served with such order, he shall be guilty of a Class 1 misdemeanor.
If that is the only relevant provision, then the answer is “yes.” It’s not criminal trespassing unless there is a sign posted or the person has been told in person to leave.
There exists also civil trespassing, which wouldn’t be covered by this provision. But the remedy for civil trespassing is a lawsuit.
There’s nothing particularly surprising about either of these things. In general, the law doesn’t want to create problems. Assuming that all unauthorized stepping on someone else’s property without something further, like a breach of the peace, is a violation of law would create unnecessary legal conflicts. One would prefer that general peaceful activity, like just walking around or playing soccer, not constantly inadvertently create breaches of the law.