The data you link to shows that the top 5% income people pay 58.55% of all taxes, while the the top 50% pay 97.22%, as you indicated.
So as a simple-language summary we might all agree that the rich pay a majority of the taxes, while the richer half of the country pays nearly all the taxes.
Note that the "richer half" starts at the median income. For the US, the median income is about $52k. That isn't particularly wealthy. When the bar for "nearly all the wealth" is set at that point, it is not surprising at all.
http://taxfoundation.org/article/summary-latest-federal-inco... Shows the top 50% paying nearly all the taxes. That is a far cry from the rich.