Off the top of my head, for Lincoln...
The only legal requirement for notification is regarding CHP holders that are actively carrying must declare during any "official" contact with LEO or EMT.
It would just be stupid not to notify if open carry is involved. Locked in the trunk, I would not bring it up unless asked directly
Don't forget, Lincoln has a ridiculous ordinance that makes it illegal to store a firearm in a vehicle for more than 24 hours at a time.
Plus Lincoln has some other ordinances, unique to the city, that could turn conviction of certain misdemeanor crimes into a felony level firearms possession ban. Make sure you are not prohibited
Nebraska law does not define loaded or unloaded firearms. No requirements to unload the handgun or rifle that I know of, but shotguns can not be loaded or even have that magazine charged while on/in any motor vehicle ( game laws)
The Nebraska's definition of Concealed handgun means the handgun is totally hidden from view. If any part of the handgun is capable of being seen, it is not a concealed handgun. On the street, that definition may not fly far, so err on the side of caution