A tax refund is based on paying too much taxes in the first place.
When you are single and working only one job, claiming 0 on your W-4 form means that you are paying more taxes than you need to pay and you would get a refund of the over payment.
When you change to 1 allowance, less taxes are taken. You wouldn't be overpaying by as much and your refund would go down, but it is still likely that you would get a small refund.
When you change to 2 allowances, but you are only a single person, you are right on the edge of not paying enough taxes. You would either break even or you would end up owing a small amount. So - either a very small refund or a very small tax debt to the IRS.
Now, this year is half over and you have been claiming 0 for the first six months of the year. That means for this year, you could probably claim 2 allowances for the rest of the year and be fine.
During either December or January, change it to 1 for the following year if you want to avoid owing taxes at the end of the following year.