Do ND filters increase the dynamic range that a camera can 'see'?

It's just dark glass (or resin, of course); it reduces the amount of light entering the lens, and that's all. There may be a slight amount of nonlinearity in the colour response (causing a slight colour shift) but that is merely due to the absence of real-world, uniformly attenuative substances.

A graduated (or split) neutral density filter can be used to control contrast by selectively (by position) controlling which parts of the image are viewed through the filter, but a solid ND is just an attenuator.

Your question references ND filters which I think most would consider to be a solid ND filter. Those are mostly used to decrease your shutter speed for long exposures or by allowing you to open up your aperture for a more shallow depth of field in bright light.

A graduated ND filter is typically used to balance the exposure. With a traditional ND filter the top part of the filter is darker (by the rated stops for the filter) and it gradually moves to transparent. You place the darker part over the brightest parts of the scene (a bright sky) in order to properly expose the foreground.

In the foreground you do not lose shadow detail because it's viewed through the transparent part of the filter. If anything you gain shadow detail as you're able to properly expose it without the bright sky causing the foreground to underexpose.

However, if you have shadow detail you need preserved in the bright area...those would most likely be lost anyway due to the bright source and just about your only choice would be to use an HDR technique. Some of the highlights would be lost in the bright area but I think if you're trying to balance the overall exposure this is a point of choice - what do you want to show in the image? Before filters or HDR you always had to choose - expose for foreground or expose for the sky/background.

There are also reverse grad ND filters which help with sunsets/sunrises - when the brightest part of the sky is along the horizon line. You are able to preserve highlight and shadow detail in the sky above the sun due to how the graduation is applied to this filter. It's darkest at the center, bottom is transparent...but from the middle dark area it gradually lightens toward the top.

Both your graphs are correct.

If you represent the intensity levels in a linear scale, then the left graph (“Linear mapping”) is the appropriate one. If you represent intensity levels in a log scale (i.e. in number of stops, or EVs/LVs...), then the right graph is appropriate.

BTW, the filter does not clip darks. The “ND filter” curve on the right graph can be extended to the left, to negative intensity levels (negative EV values are OK).

You use the ND to get to your ideal settings.

If you digital sensor has max dynamic at 800 ISO but this would overexpose the scene you use an ND to compensate.

So in this case you could say you gain dynamic using an ND opposite to reducing it in camera.