Lou,
I try to avoid single element diopters as I prefer not to add any extra optics unless absolutely necessary. However, for the sake of the experiment I suggested, you need to pick a diopter of a strength that will allow you to focus on a subject the same distance away, with and without the diopter and with and without the dual element close-up lens.
As a hypothetical example, put the camera and lens on a tripod. Without the close-up device move the camera and lens to near the minimum focus distance of the lens. Look carefully at the size a subject (ideally a ruler or measuring device of some sort) in the frame. The keeping the distance from the subject the same, add the close-up device. Refocus (the lens scale will indicate a greater distance as the close-up device is altering this) and look at the size of the subject again. If it is the same size or fairly close to the same, the close-up device is PROBABLY not making a significant difference in the angle of coverage of the lens.
BUT, depending on the lens itself, the angle of coverage of the lens may change a bit as the lens is focused closer and closer. This is particularly true with IF (internal focus) lenses which actually shorten the focal length a little as they shift the elements to change focus without extending the whole optic outward in the way a helical focus ring would do. This can also be verified by putting the lens and camera on a tripod. Shoot a frame with the lens focused at infinity at an aperture about f11 to f16. Reshoot the same shot with the lens focused closer and the same (good depth-of-field) aperture. Compare the two shots. In my experience with internal focus lenses like Nikon's 17-35mm and the newer 14-24mm, I can see the frame get a little (not a lot) wider when the focus is closer. I can see this just be pressing the depth-of-field preview button.
With the older 105mm Micro-nikkor, the close focus is achieved by moving the whole set of optics outward from the film or sensor. And the angle of coverage should not change significantly. This would be an ideal lens to try the various type of close-up devices with. An older manual focus wide-angle, such at the old 28mm f2.8 would also probably be similar in this respect. Newer lenses, particularly the newest wide-angle zoom lenses, have very complex movements of their elements relative to each other.
Fred