In your scenario, it might be a "simple assault." In real life, it is not that simple. First question: Why were the police not called to the scene? If it was at a convenience store, there might be a video record.
Either way, whether it is a patrol unit or the victim going to the station, at least the following information should be gathered: 1. Name, address, phone number, contact information from victim and any witnesses. 2. Written, signed, witnessed statements from all parties involved, to include circumstances leading up to the assault. 3. Documentation/photographs of any medical treatment and/or diagnosis. 4. Reporting officer will document details like weather, time, name/address/description of perpetrator (since victim knows who it is) If they did not, they would need things to identify when/if they caught him/her. That would involve things like vehicle identification, direction of travel, identifying features.
Those are things needed for the basic report........then, depending on the location, size of department and other factors.
(a) If the officer thinks there is probable cause, he would go before a judge (usually a magistrate) and swear out a warrant for that person's arrest. or.....
(b) If the officer or the judge thinks there is more to the story than they are hearing, they might decline and tell the "victim" their next steps. The old "extenuating circumstances," might be showing. You rarely get the complete story from anyone.
It comes down to evidence that can be put in front of a judge. I have had hundreds of people call and say, "John Doe is doing this and this and I want you to lock him up!"
I have to explain that it isn't done that way. The person making the claim has to be willing to stand in front of a judge and swear under oath that what they claim happened really did happen. That an anonymous telephone call means very little when it comes to taking away someone's freedom. That they have to be willing to get involved. 95% of the time, I never hear from them again.
So to answer your question, after all that background, what happens depends on what you intend to happen, but the actual details after the report depends on whether the victim and witness are willing to testify. If there are no curlicues, then the policeman would get a warrant and normal arrest, booking, arraignment, plea or trial, probation would follow.
Here is how I define the possible curlicue: Person B just happened to walk by the store while "victim" and witness were stealing from the store and Person B witnessed it. Person A cuts themselves, smears blood on their face and goes to police station and makes report to deflect attention from the in-house robbery. Not saying it is likely........but I have actually seen/dealt with this type scenario. Person A wound up with "false report of a crime" charge.
Trust me on this, it really happens. I had a guy wind up in jail for a year on a stalking charge. I didn't then and don't now think he did it, but his lawyer could not come up with anything to disprove the case........And if you're wondering, I told the judge of my doubts, but it did not do any good. When he got out, I told him to keep a journal of where he went, so he could prove where he was when, so he did not get in the same situation again.