I'm no economist, so take this with appropriate skepticism:
I think the issue is that there probably isn't a "true" inflation. The main contention seems to be between those who use inflation as a measure of cost of living, and those who use it as a measure of policy-driven change in the value of money. Both views have their uses. Personally, I prefer the growth in money supply view because it seems to be less susceptible to manipulation by an agenda. The basket of goods is by necessity a small sample of the total cost of living, and I think that the main complaint lodged against it is that the goods are chosen and changed over time in order to portray a picture that's better than it would otherwise be. shadowstats.com has more information on how the CPI changes with selections in the basket of goods.
Using the money supply as the definition of inflation gets closer to the viewpoint I care about, which is trying to get a sense of how policy is changing the value of my savings and income. I like to look at a simple ratio of money supply growth to GDP growth for some idea of how policy is changing the value of money, but this too is fraught with danger because it ignores important things like technical innovation, changing resource bases, and money velocity. So, I think it's a matter of personal preference and probably the best policy is to look at it from as many angles as possible if it's something that's important for your decision making.
That's a very fascinating link. Are the claims made here true, that these charts are true to the 1980 definition of inflation? Are there any other factors that could make what is in these charts incorrect or misrepresented? I'm not from the US and don't have a first-hand impression of US price development.
It seems unlikely to me that the US has really had >7% yearly inflation for well over a decade. So what gives? The most compelling counterargument is simply that the charts in your link don't reflect reality.
So the inflation calculations might indeed be inaccurate but there isn't much solid data to back it up.
I personally find it better to focus on issues that have lots of data such as the debt crisis , the growing costs of SS and medicare and the harmful effects of industry/government collusion  which are all very real and can be proven.
Usually it involves taking one common and fixed item, like cereal, and looking at its prices over time. While there are seasonal effects, looking at the price over a number of years gives you a good picture as to the actual inflation.
The key thing here, though, is "fixed". Sometimes companies will make adjustments to the product to reduce costs, and those adjustments make the apparent inflation appear lower than they really are. For example, a jeweler can quote the same price for jewelry next year (assuming gold price rises) if they use less gold and more base metals in the jewelry. CPI measure would say that there was no change, although for all intents and purposes there was a change.