You are correct. Where I live, I am surrounded by National Forest with four National Parks nearby. I should have made it clearer that it is legal to carry a loaded handgun in a National Forest, as long as your state allows it.
Most people would be shocked to learn that it is perfectly legal to carry a loaded handgun in a National Forest in California (don't carry a rifle though, unless hunting it is season).
Right... they're a little touchier about rifles, because they're usually combatting poachers hunting out of season, or hunting from their vehicle.