The reason you don't see any spots on the bright white image is two-fold:
- Light is entering your camera at an aperture size equal to the throat diameter of the Nikon F-mount. That is larger than your camera's sensor. The dust spots on your camera's sensor stack are not visible because unfocused light is coming from around the edges at all angles and there are no shadows being projected onto the sensor below the glass cover.
- The exposure is so bright that the entire image is at maximum saturation/pure white. Any details that might have been visible at a lower exposure level are completely blown out by the overexposure.
Your example is exactly what one would expect with something on your sensor (dust or spots of dried liquid, etc.) when you take a photo with no lens on and the exposure completely blown out.
On the other hand, it takes a fairly large chunk of something on a lens surface to show up in a photo, and they are much more diffuse and larger than spots caused by material very close to the sensor.
The second image looks very much like there are drops of some type of oil that has gotten on your sensor and then small pieces of dust might have gotten stuck to the oil. Nikons have a reputation of coming from the factory with a little too much oil on the shutter mechanisms. Some of the excess oil eventually finds its way onto the sensor. The fact that the spots in your image are very uniform in size, shape, and opacity strongly suggest drops of some kind of liquid, or bits of dust stuck to such drops, are the source of the spots.
In the case of your D80, it's probably safe to say it is not a factory fresh camera. However, if you have recently had some service work done that included oiling the mirror or shutter mechanisms it might explain the recent appearance of oil on your sensor. Another possibility is that the spots have been there a good while and you just recently shot under the conditions that cause them to be most noticeable (narrow aperture, relatively dim exposure). Or it may just be another type of wet substance that has gotten stuck on the front of your sensor stack.