I've tested it. I wore polarized sunglasses and looked at a car. The reflection of the sun did not disappear.
I wouldn't expect it to, with polarized sunglasses.
I'm not sure why you are looking for something to disappear or not disappear, but the difference with polarized filtering and full-spectrum light filtering means that you shouldn't expect one to perform the same as the other.
I guarantee you that if you see a reflection of sunlight on a car's surface, using a solar filter will make it completely disappear, along the car and everything else. It will be completely dark.
If you see a reflection of the sun in the windshield, unlike a polarized sunglass lens, a solar filter will knock out everything EXCEPT a small circle of the sun...the actual sun; not the glare, flare and blooming of light that appears around an unfiltered sun. Your polarized filter won't do that. It's not supposed to, which is why you are confused in equating polarization with light filtering.
In a little while, I'll go out and take photos of filtered and unfiltered reflected sun and show you what I mean.
Per your second question, Rowbotham provides an image in Earth Not a Globe.

In the above image the sun is projected on 6, 7, 8, 9 and 10.
One can also imagine the screen of the atmosphere as a "dome" around the observer rather than a plane.
Like the Bat Signal projecting onto clouds or haze in the sky?
Or like a movie being projected onto a screen from behind the screen?
The car reflection model/illustration you proposed is like the former.
ENaG suggests to me the latter.
Which is it?