I don't think you need to put it in your application however I would suggest that you definitely mention it in the interview - so as to legally cover yourself - verbatim is considered legal. Instead...
I definitely wouldn't quit over that. You have worked hard to get where you are and deserve to give yourself a desent chance at this career! If I was in this situation, despite my ego, I would...
Totally sympathise with your situation.....has happened to me before. Maybe you could look at being more assertive with your colleagues. Always ask for help when you need it....we are suppossed to...