Consider getting a dashboard DVR camera. There are hundreds of them out there for as cheap as $50 or so. They come on when they have power and record continuously. There won’t be a he-said-she-said situation when you can show the whole thing on video.
Your insurance company will hire you an attorney if you get sued. Go with your phone and take videos of the intersection from all angles showing the stoplights and how they change.
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Your insurance is who will be dealing with lawyers, should they be needed.
The terms of your policy assign all rights and responsibilities to the insurance company.
I litigation arises, your insurance company will do the legal stuff.
You should talk to ONLY YOUR INSURANCE COMPANY - chatting with others (especially the other driver) is like talking to the police when you are a suspect - there is nothing you can say which will help, but you can hurt your position.
Reported.