I don't know where you getting your information off of the handgunlaw.us website but I hope you are wrong as I take firearms with me into Oklahoma regularly. The handgunlaw.us is pretty clear and includes the citations for Oklahoma law.
21 O.S. Section 1290.26 (Effective Date of This New Law 11/1/12 as Amended by SB 1785)
A. Any person entering this state in possession of a firearm authorized for concealed carry upon the authority and license of another state is authorized to continue to carry a concealed firearm and license in this state; provided the license from the other state remains valid.
Carrying a unloaded handgun concealed doesn't make much sense does it?
Title 21. Crimes and Punishments Chapter 53 - Manufacture, Sale, and Wearing of Weapons Oklahoma Self-Defense Act Section 1290.2 – Definitions as Used In Sections 1 Through 25 of This Act:
1. "Concealed handgun" means a loaded or unloaded pistol carried hidden from the detection and view of another person either upon or about the person, in a purse or other container belonging to the person, or in a vehicle which is operated by the person or in which the person is riding as a passenger
§21-1289.7. Firearms in Vehicles.
FIREARMS IN VEHICLES
Any person, except a convicted felon, may transport in a motor vehicle a rifle or shotgun concealed behind a seat of the vehicle or within the interior of the vehicle provided the rifle or shotgun is not clip, magazine or chamber loaded. The authority to transport a clip or magazine loaded rifle or shotgun shall be pursuant to Section 1289.13 of this title.
TRANSPORTING A LOADED FIREARM
Except as otherwise provided by the provisions of the Oklahoma Self-Defense Act or another provision of law, it shall be unlawful to transport a loaded pistol, rifle or shotgun in a land borne motor vehicle over a public highway or roadway. However, a rifle or shotgun may be transported clip or magazine loaded and not chamber loaded when transported in an exterior locked compartment of the vehicle or trunk of the vehicle or in the interior compartment of the vehicle...
I see other THR members posted while I was writing my response. Well I guess we covered it well.
So you are ok with concealing your unloaded hunting gun in the passenger compartment.
Loaded guns need to go into the trunk or exterior locked compartment.