They did not "put words in his mouth" but they did miss out almost all of what he actually said, which did indeed change the perception. When he concluded with "fight" it was in the context of what he had previously been saying about 'fighting' the injustice of a "stolen" election, and encouraging the crowd to persuade Congress to overturn the result. The perception the BBC presented implied he was precipitating the violence that followed.
Yes, it was likely that a certain amount of unrest might have been anticipated, but Trump did not demand that the Capitol be stormed.