A minor interdomestic dispute (as the police would see it) - do you really think the police would pay a visit? You might be right, of course, but I doubt it. You're spot on about the CPS - I was trying to keep it simpler than talking about discretionary disposal powers!
There really isn't any such thing as putting it "on the record", nor do you decide whether to press criminal charges or not. You can go to the police to report a crime, if you believe that a crime has been committed (from what you say, perhaps assault?). The police will then decide whether to...