The pictures on the D800 are better than when I've used it on the D90 because the D800 has a 36Mp full frame sensor and the D90 has a 12Mp crop frame sensor. In postproduction, if I crop the frame from a 300mm FX lens on the D800 to the 450mm equivalent of a DX lens mounted on the D800, the image area on the sensor will be about 20-24Mp. I can, therefore make larger enlargements when the lens is on the D800. I can crop the pictures to an angle of view that substantially exceeds that of a 500-600mm or more lens in postproduction and maintain better or equal quality to the same lens on the D90.

The same thing is true for other lenses. For example, I have a 24-70mm f2.8 FX Nikon lens that I keep on the D800 95% of the time. Because of the D800's high megapixel sensor, I can take a picture with the lens at 70mm and crop the frame in post production so the angle of view is that of an approximately 200-300mm lens without an appreciable loss of image quality.

I think you have it wrong. The lens in question is a DX (crop) lens, not an FX (full frame) lens, and its real focal length is 70-300mm, with an image circle that is just sufficient to enclose a DX sensor. If you were to use it on a full frame camera that does not automatically limit the portion of its sensor used to that of a crop sensor, the resulting picture will be a circle, not a rectangle because the image circle of a DX lens is smaller than that of an FX lens.

If you use an FX lens on a DX camera, the sensor will crop the image to cover the same angle of view as a 105mm to 450mm lens. That is because the image circle on a full frame lens is substantially larger than the DX sensor's rectangular shape.

In any case, the focal length stamped on the lens is the true focal length regardless of size of the camera's sensor it is used on. If the lens is 300mm, it means that it will bring an object at infinity distance into focus when the rear of the lens is 300mm from the sensor regardless of the size of the sensor. A 300mm FX lens does not change its focal length to 450mm when it is used on a DX camera, it crops the image to the equivalent of 450mm because the sensor utilizes less of the area within the image circle. Therefore, a 70mm-300mm DX lens is 70-300mm on a DX camera and not the equivalent of 105mm-450mm as the vendor indicated in the add.

In reality, I think I have it right - a DX camera is a CROP sensor camera, so the field of view is multiplied by the crop factor, on a Nikon that is 1.5, so that gives ANY lens the equivalent field of view of a lens that is 1.5 times longer than on an FX camera. The field of view is often mistaken for magnification, it is NOT magnification though.

I agree 100%, a 300 mm lens IS a 300mm lens regardless of the crop factor, BUT, the FIELD OF VIEW is multiplied by the crop factor and equates to the same field of view of a longer lens on a FX (non-crop) camera.

In reality, I think I have it right - a DX camera is a CROP sensor camera, so the field of view is multiplied by the crop factor, on a Nikon that is 1.5, so that gives ANY lens the equivalent field of view of a lens that is 1.5 times longer than on an FX camera. The field of view is often mistaken for magnification, it is NOT magnification though.

I agree 100%, a 300 mm lens IS a 300mm lens regardless of the crop factor, BUT, the FIELD OF VIEW is multiplied by the crop factor and equates to the same field of view of a longer lens on a FX (non-crop) camera.

No, not so, it is the sensor that determines the crop factor - can you show some proof of your conclusions?

If I use an FX lens on my Nikon, it will be a cropped field of view, if I use a DX lens it also is a cropped field of view.... You have an issue if you use a DX lens on an FX camera because of the image circle, but not if you use an FX lens on a DX (The sensor sits in the middle of the circle)

On a Canon, the crop factor is 1.6 (I believe), again, the sensor determines the crop factor, not the lens.

https://photographylife.com/nikon-dx-vs-fx/

The sensor is only half of the equation. The lens is the other half. The crop factor is a combination of the two.

Here is a method for proving it yourself. Take one DX lens and one FX lens, the focal length is not important. Use a white sheet of paper as a target. Take the lenses outside and first take the DX lens and holding it in you hand move it back and forth using the white paper in place of the sensor or film. Adjust the distance of the lens to the paper so that a subject at infinity is in focus and note the diameter of the circular image on the paper. Now repeat the same process with the FX lens and note that the circular image is larger than that of the DX lens. A camera's sensor size determines how much of that circular image is in your picture. The minimum image circle for an FX lens (sensor size about 24 X 36mm) is approximately 44mm; i.e., the diagonal measurement of the sensor. The minimum image circle for a DX lens (sensor size about 24 X 16mm) is approximately 29mm.

For a DX lens on a DX body the crop factor is always 1:1. For an FX lens on an FX body, the crop factor is always 1:1. When a DX lens is used on a FX body, the camera's electronics uses only the center part of the sensor (24 X 16mm) since anything outside that area is useless information. It is all mathematics.