The details are subtle, but you are very close to the whole picture.
18.2-302.1 is the school carry law. There are two parts of it germane to this question.
The older part is the "shall not apply to" sections in the paragraph after paragraph C:
The restriction set forth in this law "shall not apply to" "(vii) a person who has a valid concealed handgun permit and possesses a concealed handgun
while in a motor vehicle in a parking lot, traffic circle, or other means of vehicular ingress or egress to the school."
This is where you got the "must stay in the car" idea, because that is what this "shall not apply" provision says.
However, earlier in the code, in that same paragraph we have this:
"The exemptions set out in § 18.2-308 shall apply, mutatis mutandis, to the provisions of this section."
That means that the list of exemptions in the CHP section of code also apply to this section of code. Last year, it was this list of exemptions that was updated to include the storage of a loaded firearm in a secured console.
So... to sum up... there are two provisions for carrying a loaded firearm onto school property. The older one says you must stay in your car, and is designed to apply to parents dropping off or picking up kids. The newer one essentially says it doesn't count as a firearm on school property if it is in a secured container, with no restriction on whether you park, get out, etc.
Hope that helps!
TFred