The more I've thought about my question about no-trespassing signs saying "POSTED", I've realized that "no trespassing" doesn't really make sense either. Trespassing is, by definition, illegal, making "no trespassing" a tautology. Saying "no trespassing" is equivalent to putting a "no stealing" sign on a car or a "no speeding" sign on a road. "No trespassing" is a true statement anywhere, even if the public is allowed, because if everyone is allowed to be on the property, nobody is trespassing, making "no trespassing" an irrelevant statement, just as a "no stealing" sign on a car is irrelevant to someone who is not stealing it and a "no speeding" sign is irrelevant to someone who is driving at the speed limit.
Why do some private property signs say "no trespassing"? "Private Property" seems like the most logical wording in cases where "POSTED" is not legally necessary, and while it is common, "no trespassing" is also quite common.