96% of donations went to the Clinton Family Foundation, not the Clinton Foundation, and this is an important distinction. Both of the articles you link carry corrections to this effect, but this article goes into more detail about why it matters in the context of a Wall Street Journal article carrying the same claim (I can't verify whether the WSJ have issued a correction, because the article is behind a paywall).
The organisation commonly called the Clinton Foundation, officially the Bill, Hillary & Chelsea Clinton Foundation, is a charity in its own right, with many donors. It largely carries out initiatives independent of any other charity. Its own FAQs go into more detail. It has sometimes been the subject of controversy (many charities are), but it is not the subject of this particular "revelation".
The Clinton Family Foundation, on the other hand, is a clearing house, which organises most of the donations the Clintons make on their behalf. This is not unusual, and can be thought of as like having a separate bank account for money you plan to donate. The actual money donated to the Clinton Family Foundation is then donated, on behalf of the Clintons, to many different charities. It is not at all surprising to find that it received 96% of her donations, since that is its purpose — to act as a clearing house for donating them to the appropriate charities.
According to the Daily Caller article you cite, in 2014 the Clinton Family Foundation donated — on behalf of the Clinton family — $1.87 million to the Clinton Foundation, out of a total of $3 million. If we take that proportion as representative of how much Hillary gave, it amounts to roughly 60%, not 96%.
To the second question, "could you consider this to be 96% (or even 60%) that did not go to charity?", the answer is a simple "no", since the Clinton Foundation is recognised as a charity.
Regarding the claim that she donated to "a charity she controls", the Clinton Family Foundation is controlled entirely by the Clinton family (this form linked in Abedin's answer lists Bill as President, Hillary as Secretary/Treasurer, and Chelsea as Director). Again, this is in keeping with its role as a vehicle to manage their donations. They are also on the board of the Clinton Foundation but do not have sole control. So they controlled the fact that the Clinton Family Foundation passed on 60% of their donations to the Clinton Foundation, and have some say in what the Clinton Foundation does with that (and other people's) money, within its established framework.