I live in Texas. The law in plain English is this. You can carry a handgun in your car while driving anywhere. That is any public place, except where specifically prohibited.(school grounds.etc) It must be concealed. You do not need a license for this. When you leave your vehicle, to carry the gun ON YOUR PERSON out of the vehicle, then you need to be a licensed permit holder.
Of course you must legally own the gun, and have a clean record, etc. I know it is a little confusing. I think they came up with the gun in the car law because of carjackings. I hope this helps.