You quit. The answer to your question is it doesn't matter. You will eventually learn that almost all jobs have issues like this and you may perceive some rules as "unfair" or that other people are being favored, but that's life. You already know the way you quit wasn't the way to do it. Also, many part-time people get NO benefits, so that is something you'll learn also.

I wouldn't use the job as a reference for getting other jobs, because you won't get a good reference from them.