The term DX on the 35mm AF-S f/1.8 indicates the lens is designed for a crop sensor format. On crop sensor cameras, 35mm is not considered wide angle. They (DX lenses) don't produce an image circle large enough to cover a full frame camera. When used on some full frame cameras - they will crop the center out and you effectively still get the same view as a crop sensor camera.
The 35mm AF is not a crop sensor lens and will work on both crop sensor cameras and full frame cameras - although on a full frame camera it will have wider field of view than on a crop sensor camera. Because of this wider field of view, it is considered somewhat wide on a full frame camera.
On your D7000, they will both have the same field of view (not wide, more of the 'normal' range, around the same field of view as a 50mm lens when mounted on a full frame camera). Imagine two circles (two image cirlces projected by their respective lenses), one bigger than the other (the DX being the small circle), but the same size rectangle (your sensor) in the middle of each - just large enough to fit fully in the small circle. They're not magnifying the image more than each other (since they're the same focal length), so the rectangle in the middle is still representing the same portion of the image.
On a full frame camera, leave the two circles the same, but now the rectangle (sensor) is bigger than the DX circle.
The answer lies right in the naming of the lenses. The Nikon 35mm f/2D AF Wide-Angle Lens is designed for full frame cameras(but it can be used on any Nikon DSLR). The Nikon 35mm f/1.8G AF-S DX lens is designed for crop sensor or smaller then full frame cameras. What this means is that the second lens is not going to work on a full frame camera, rather it only works on a crop sensor camera.
The important piece to note is that your D7000 is a crop sensor camera, so both the DX and non DX lenses will have the field of view of a 52.5mm lens on a full frame camera due to the 1.5x crop factor. So you will not be able to use either as a wide angle lens!
To sum it up, the Nikon 35mm f/2D AF Wide-Angle Lens can be used on a full frame camera, as a wide angle, the Nikon 35mm f/1.8G AF-S DX lens cannot - thus it is not a wide angle lens.
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Probably the simplest way to compare lenses across different sensor sizes is to normalize everything as if they all worked with full-frame sensors. In a normalized context, its a lot easier to compare lens behavior. In the case of full-frame sensors, the following rough scale for angle of view exists:
The Nikon 35mm f/1.8G AF-S DX Lens is designed to work with APS-C, or cropped, sensors. As such, it's image circle is smaller than a Nikon 35mm f/2D AF Wide-Angle Lens which was designed for a full-frame sensor. The difference in image circle sizes is realized as a difference in angle of view. Nikon APS-C sensors have a 1.5x "crop factor" in relation to their Full Frame sensors. To easily compare the angle of view for a cropped sensor lens with that of a full-frame lens, simply multiply the focal length of the cropped sensor lens by 1.5:
The 35mm DX lens "behaves like" a 52.5mm full-frame lens from an angle-of-view perspective. Since 50mm lenses fall within that "normal angle" range rather than "wide-angle" range, it wouldn't really be appropriate to call it a wide angle lens. On the flip side...if you attached the 35mm full frame lens to a DX body, it would also behave like a 52.5mm lens, and also no longer be wide angle.