What looks worse on a resume: quitting or being fired?

(note: sorry this is so long. TL;DR at the bottom)

About six months ago, I started my first part-time job (also my first job, period). Part of our duties was selling discount cards (there was a quota) and I constantly struggled because I was too shy to be forceful, and I’d rather the customers be happy with me than push past the first no.

I’m about to go off to college over 500 miles away, and I had a day a couple of weeks from now when I would officially stop working. But then I got a call from my manager saying two coworkers had approached her two separate times to tell her that I said “I don’t care about selling cards since I’m going off to college so soon anyway.” I know for a fact that I never said those words. Maybe I said something about not caring about my recent failure, trying to imply that I wouldn’t let it get in my head and mess with my future performance? But I would never outright say that I didn’t care about cards. I was so bad at it, they were literally all I thought about. I worried about them constantly at work, and often at home.

Because of what I allegedly said, my manager gave away my last two weeks’ worth of shifts. I never even got a chance to plead my case. I don’t know if in work you get a chance like that.

I’m going to go in soon to clean out my locker, but I’m wondering if I should just quit and cut all ties. In the future, when applying for a job, would I have to explain why I quit? It’s partially because of school, because I wouldn’t have time to work over breaks, and I’d kinda rather sever all ties before going to school, and when next summer comes around, I can see where I stand, whether I want to reapply or not. Can I say that was my reason for quitting and totally omit the drama?

Also, does it look better on a resume to have quit, or to have been canned? I don’t know for sure if I’m fired or not, but if my manager is thinking about it, would it be better for me to quit? How do references even work when fired vs. when quit? Since it was my very first job, and only part-time for six months, can I put it all behind me and not have future employers look too closely at it? I know I’ll want another part-time job when I’m at college, so will it come up and be an issue then?

I’d ask my mom, but she’s going crazy with Mama Bear instincts and talking about boycotting the store, so I don’t think her judgment is best right now.

Sorry for the length. TL;DR: Things are ending badly at my first part-time job, and I’m wondering if it’s better for me to wait and see if I’m fired, or to quit, in respects to future plans.