frontal area is the surface area presented to the wind or air by your hood, front cowl, front headlight assembly or fascia, intercooler if you have one or bottom part of your radiator , and yes, the front air dam.
if you lower the car, you haven't changed the surface area. you still have the same parts that are smacking up against the wind as the car goes forward.
lowering the car is mainly to help change the center of gravity's location to a better or more stable location for cornering and front to back weight transfer as you corner or accelerate in a straight line,
what is frontal area?:
Imagine if you had a brick wall and were pushing it from behind with your car.
The area (height x width) of the brick wall is what is generating resistance against the oncoming wind or air.
the wind or air smacks up against that wall surface and pushes against it. That's what is slowing you down (partially).
the smaller the brick wall, the less you have to push.
now, back to topic on the effect of lowering the car and drag resistance:
by lowering the car, you decrease the amount of air that goes underneath the car. this air travelling underneath can generate turbulence under the car and turbulence as it exits from the rear of the car by creating vortices behind the rear bumper.
turbulence adds drag.
but the lowering effect has no bearing on area. you're just making air flow over the car more rather than some getting underneath...which is in itself a good thing. You do reduce the drag a bit that way.
Those wide mouth air dams add front surface area. A lip is not a huge change in area. A lip prevents air flow from getting under the car. That's why it's good. You should read the threads here that talk about what a front air dam is supposed to do and what a rear splitter is. The front air dam works together with a rear splitter to increase downforce and reduce drag.
PS
here is the article I wrote back in March 2002 about air flow control under the car for increasing downforce in a straight line