Q: The Act of Agriculture states in article 22, that "Out of towns and villages one must contact the landowner or the respective authority before using campers, camping cars, caravans etc outside of designated campgrounds." So, my question is: "Is sleeping in a regular car legally allowed outside a campground"?
A: Parking vehicles on private land without permission is not allowed, and that is regardless of if it is a campervan or a small car.
The part about "the respective authority," however, refers to agencies like local municipalities, national park authorities or the roads administration for public land. In these cases the quesion becomes one of making a definition bwtween 'camping' and 'taking a nap.'
Roads around Iceland have rest areas for people to pull off at, and it is actively encouraged to take a nap if you are tired while driving. How long can you stay there before a nap becomes 'staying the night'? There is no clear answer to this.
However, you are not allowed to set up tents in public areas outside of campsites, and it is certainly not okay to leave litter behind, or tend to the "call of nature."
In short, you are unlikely to get in any trouble for sleeping in a small car in a road lay-by, but anyone traveling around the country for more than a couple of days would be well-advised to use campsites, because they are cheap, comfortable, and have welcome amenities like drinking water, showers and washing facilities.