Did Western media and human rights organizations lie about the situation in Sanjar to trigger US military intervention?

So now we know that Western media lied about the plight and size of the refugees in Sanjar:  "Administration officials said that several thousand Yazidis remained on the mountain, not the tens of thousands who originally were there. Some of the people on Mount Sinjar indicated to American forces that they considered the mountain to be a place of refuge and a home and did not want to leave, a second United States official said. Both officials spoke on the condition of anonymity because they were not authorized to speak publicly on the issue...The speed with which the Obama administration announced that the siege had been broken may cause some consternation overseas, given the increasingly dire descriptions from aid agencies about the crisis on Mount Sinjar."