Good for you on not just giving up and caving into their tactics. Too bad they continue to make it difficult for you.
No, you are not obligated to pay anything. However, that doesn't mean they couldn't continue to make life difficult for you. I'm assuming the "collection notice" you got is from Budget, and not actually a collection agency. Unless they take you to court and successfully obtain a judgment against you, you are free to disregard any notices from them.
Since I wouldn't want this hanging out there, here's what I would do (and it always works for me): I'd call the corporate office, either the head of the company or the legal department. You won't get to actually talk to the CEO, of course, they'll have someone else talk to you instead. Tell them politely but firmly that they need to cancel this bill and stop sending you bills or notices. If you don't get satisfaction from whomever you talk to, demand to speak to the next person up in the hierarchy. Keep going until you find the person who makes it go away.
Whenever I've had situations like this, I find that it usually takes around 20-60 minutes on the phone to finally make it go away. Try real hard to enlist everyone you talk to as your friend and ally, and resist the urge to start yelling at the new person even though it's frustrating. What works much better is to say cheerfully, "Hi, I'm hoping you can help me with a problem I'm having." They'll always say, "Okay, what can I do?" or "I'll do whatever I can." This immediately makes it their problem, they are now on the hook, so to speak, to solve it. It's a weird tactic I learned a long time ago, and it works really well.
If they really stonewall you, which is rare, ask to speak to the legal department. When you talk to the legal department, ask them for the address to serve a Small Claims summons against them. They'll ask, "What for?" Then you explain the problem, and they will definitely fix it for you.