These two images are identical horizontally, despite one being anamorphic and one being letterbox. Where, exactly, is this "extra" horizontal resolution that should obviously come from squeezing an image larger than 720 pixels wide into a frame that is only 720 pixels wide?
Look at the black bars at the tops and bottoms. The anamorphic picture has smaller black bars, because there is more vertical resolution (in regards to the actual picture information), as you, yourself stated. This has the effect of making the people look taller and skinnier, which looks similar to a picture that is squeezed horizontally, but in fact isn't - in this particular case.
Now, if the anamorphic picture and 4x3 picture are identical width-wise, but the vertical resolution of the anamorphic transfer makes the people look skinnier, then there are two solutions to correctly display the original aspect ratio. 1) Remove or convert 4 vertical lines into 3 throughout the picture, to reduce the vertical height [4x3 display] or 2) stretch the picture horizontally [16x9 display]. However, the image that is being stretched is the original 720 pixel wide image!
A 1920 pixel-wide image cannot be "squeezed" into a 720 pixel-wide like anamorphic film - it must be resized, resulting in lost resolution. A good analogy is this - take a large JPEG, say 100x100, and resize it to 50x50. Now, if you zoom in on the picture so that it is the same size on your screen as it originally was, you will see that much information is lost. Similarly if you resized the image down to 50x50, and then resize it again to 100x100, again, you will see that visual information is lost.
The anamorphic process does squeeze the visual information of film into a smaller space, but when the smaller frame is expanded, it is still the same picture! Nothing is lost in the process (or practically nothing.). This process cannot be duplicated in the DVD transfer process, because it would require a lossless codec. Since 720x480 is not high-definition resolution, information was definitely lost when scaling down from a 1920x1080 high-definition master, and MPEG, AC3 and DTS are all necessary lossy compression techniques for putting the entire transfer on a DVD. Therefore "anamorphic DVD transfers" are not the same as "anamorphic film transfers."
I will state this correction, however. Compared to a letterbox transfer, an anamorphic transfer does have more vertical resolution, but I did mispeak when I said it was added. It would be better to say that the vertical resolution is decreased for a straight letterbox transfer. Nonetheless, an anamorphic DVD does have more vertical resolution than a non-anamorphic one. To state that an anamorphic DVD is squeezed horizontally to fit into a 720 pixel-wide space would imply that it has greater horizontal resolution than a letterbox transfer in the same 720 pixel-wide space, and this is wholly unsupportable. The very fact that the only change made to an anamorphic transfer to make it fit into a 4x3 display is to reduce the vertical resolution shows the problem in your logic. If the image were truly "squeezed", then horizontal resolution would have to be removed as well, to fit into a 4x3 space, and this isn't the case. The horzontal resolution of the DVD image, as it is stored on the DVD, remains the same regardless. It is, in fact, the display of this image in a wide aspect ratio that makes the difference. This is why 4x3 tvs with an anamorphic mode can display an anamorphic picture: the vertical scan lines are brought closer together, but there is absolutely no change made whatsoever to the horizontal picture information! There is no "unsqueezing" necessary for the 4x3 anamorphic mode, because the picture was never "squeezed" in the first place. A 16x9 TV reproduces this effect by stretching the horizontal picture wider. Both TVs affect only one dimension of the picture in order to recreate the correct aspect ratio.