Please cite the applicable Ohio laws because in many states this is a misunderstood issue.
Ohio Revised Code 2923.126
(C)(1) Nothing in this section shall negate or restrict a rule, policy, or practice of a private employer that is not a private college, university, or other institution of higher education concerning or prohibiting the presence of firearms on the private employer’s premises or property, including motor vehicles owned by the private employer. Nothing in this section shall require a private employer of that nature to adopt a rule, policy, or practice concerning or prohibiting the presence of firearms on the private employer’s premises or property, including motor vehicles owned by the private employer.
Or in other words -- it's your employer's choice whether you can even have a firearm in your vehicle. And technically they have a right to search. As pure rumor control, I have heard of companies that want to shed employees and run 24/7 have the rule year round, but somehow make it a point to do a vehicle searches on the day before buck season. I don't have names, or facts -- that's why it's rumor control.
Now as for the gun buster signs:
I was taught at a Buckeye Firearms Seminar by an Ohio CCW instructor: If you enter a facility that has has the gunbuster photopic/icon of a gun on the door, and you enter they can get you for trespassing the first time, but the cops (and managers) generally won't push it for OC or CC. The "assume it was an accident" mentality. Just don't be an A-Hole about. There is some sort of ruling as I understand it that if they don't have the gunbuster photopic/icon on the door, but do have something on the door like "No Smoking" the next line "No Firearms Allowed" and a "No pets Allowed" you can only be warned of trespassing, but they can not file charges. This also extends to the parking lot as well. But in the case of something like a strip mall -- they can't enforce it on the parking lot, unless the everyone in the strip mall is a no carry establishment.
Also usually the cops won't worry about parking lot possession, on the property, unless you have a really f'ed up property owner or the cops are looking to charge you with anything.
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Regardless of how you carry, you should always strive to be the calmest, coolest, most composed person in the room.