The Visibone palette makes this very obvious because it puts similar colours together. It is not ideal by any means, but you have to start somewhere. There will certainly be more variance between different brands of monitor and computer gamma settings than with these few percent. So, I would say, keep the Web-safe palette as a benchmark. Choose your basic colours from there, but if you need colours that are not in that palette, don't be afraid to mix your own.
Extreme subtleties are a waste of time though.
Before you go any further, check out my monitor gamma test and see how well your monitor copes with basic colours. The worst offenders, I find, are cheap PC laptops, which are totally incapable of any kind of subtlety at the extreme light or dark ends, and even those colours change with viewing angle. Many office PCs have no colour calibration or any facility for adjustment, in other cases, people just don't know how to, or can't be bothered to set them up correctly. As a Web designer, it is your responsibility to make sure that the colours you use are as accurate as possible. Just a small section of the 'Filter Browser' palette Adobe Photoshop is undoubtedly the foremost image editing program available.