I was able to resolve it myself by shutting down the computer, but this could be a problem in public computers. If using a public computer, I recommend re opening a private window and if it has not logged you out, re start their computer.

Did you log out and did you close the Private Browsing mode window(s) before closing Firefox?

Did you ever visited this website in normal mode and have logged on?

Yes the private browsing was closed as well as the regular browser. Yes I have in the past visited the same site on regular mode and logged on. That still does not explain why it would keep me logged on if visiting in private mode, after the window was closed. It was acting as if the cookies were in effect

II used a public computer and I had an unidentified charged for something on amazon by someone that got into my account because it was saved, while using a private window. I'm glad it was just 1.99 but still very annoying, I know chrome doesn't have this problem, I guess I'll just keep using it.

It is always to be very cautious when you financial transactions on a public computer. If possible the use Clear Recent History or Forget and also check the Cookie Manager for present cookies.