So how do you answer questions in a media interview? You do not want to sound like a phony baloney politician dodging questions. On the other hand, you cannot just answer them in a straightforward, logical, and linear way either, they way you have always talked to your friends and family. That simply does not seem to work.
What you need to do is listen to the question, answer it to some level, and then focus on the messages that you care about, you have to bridge back to the messages that are your main points. However, the key point is to also answer the question that you have been asked.
You need to try and avoid repeating any negative words used by the reporter as you do not want to buy into their negative assumptions. But you also do not want to spend a lot of time rebutting the negative premises- just talk about the premises that you believe in and that will help you greatly.
You need to think about the context that anyone would think of when reading or seeing your interview. The reporter can only show the part of the interview where you are taken out of context and it does not sound good with the theme. So you have to think about all of that when giving answers to them.
You do not have to outright say that you disagree with the premise that the reporter favors, instead simply focus on the premise you agree with and let others deduce that you might not agree with their assumptions.