legally, they can't say how you left your previous job. they can only say your position and how long you worked there. that being said, no one actually knows what they really say but if you ever have reason to believe they told a future employer this, you can sue them. now they do ask you on job apps if you've been fired or disciplined at work. telling you they "might" fire or discipline you is not reason for you to believe that they "will" fire or discipline you and there's a big difference between MAY or WILL. if you've already received a formal write up, you might as well wait it out. if you haven't gotten one yet just bail now and keep yourself clean.
this is also going to sound incredibly pretentious, and i don't mean it to be, but it also really depends what kind of job you go for next. if it's a corporate job and your last job was a part time, they might use some third party service for work verification and they might never actually speak to HR. however if your current job is corporate, then i would be extremely careful not to burn a bridge. YMMV and every company is different