> There's the obvious idea that governments shouldn't be involved in telling people what they can do on private property

The government literally tells people what to do on private property all the time. See zoning rules, smoking bans in restaurants, alcohol sales to minors, food safety laws, animal cruelty... the list is almost endless. I can't even fathom the kind of thinking that suggests that laws end on a property boundary.

> Are we going to have a whole system of definitions of what is billboard

Yes. It's not that hard. We encode complex things in regulations all the time.