It is a very interesting question and I don't believe you have gotten a correct or complete answer. Mr elvis is not correct in that the brightness is NOT the same as long as you are within the image circle. An image circle is a circle with an acceptable amount of light at the edges.
Tim, there are two coverage concepts floating around. One defines the circle covered as the circle within which image quality is satisfactory. The other defines it as the circle satisfactorily illuminated. One would think that the second definition is laxer than the first, i.e., defines a larger circle, but this isn't always true.
The first concept is used by lens manufacturers and exacting photographers. It is conditioned on aperture; the image circle typically gets larger as the lens is stopped down. This because some off-axis aberrations get smaller, i.e., image quality off-axis improves. as the lens is stopped down.
The second one makes little sense. Two problems with it.
First, it ignores image quality in (typically) the corners. This matters for some photographers, not for others.
Secondly, there's a rule of thumb about decline of illumination at the film plane as the distance off-axis increases. The rule, which many,
not all, lenses follow, is that illumination declines with cos^4(the angle off-axis). If you do the arithmetic, you'll find that old-school extreme wide angle lenses have large falloffs. At 45 degrees off-axis, this means that illumination is two stops down from the center. The Goerz Hypergon (with star) has claimed coverage (image quality concept) of at least 120 degrees. The sharp image it puts on film is 4 stops down at the edge. This is why we use center filters on extreme wide angle lenses; they let us get better exposure at the edges, the better to use detail that's there even though it is hard to see on the ground glass.
Some modern w/a lenses use an optical trick to improve illumination's decline to
only cos^3(the angle off-axis). A modern 120 degree lens is down
only 3 stops at the edge. Every little bit helps.