To some extent it depends upon how the issue presents procedurally.
One way it could present is that the conviction is put into a gun purchase background check database, the convicted individual disputes the denial and brings a suit in court to declare that the conviction does not qualify for the background check database, and it is decided, either on paperwork and affidavits in motion practice, or following an evidentiary hearing about the nature of the underlying conviction, before a judge, who looks at the statute and interprets it. Ultimately, as applied to the fact the court would set forth the facts that the judge believes to be relevant and make findings of fact to support a conclusion of law that it does or does not apply based upon the evidence presented.
Another way it could present is that someone sells the person who was convicted a gun without doing a background check, and then the person who purchased the gun (or the person who sold it, or both) is prosecuted criminally, and then it goes to a jury trial if that element is disputed. At that point, the jury gets a jury instruction tailored to the instruction and slightly customized to the facts of the case (e.g. inserting the names of the people alleged to be dating and reworded to be grammatical in the context of a jury instruction), and the jury decides. The jury's determination on guilt, which would haves as one element that a prior conviction was based upon a dating relationship, would be upheld if there was any evidence presented at trial upon which the jury could have reached that conclusion.
Counsel for either party could push for further clarification of anything ambiguous about the statutory language in a jury instruction, but this language is clear enough to meet a "void for vagueness" challenge to its constitutionality.
Precedents would come into play primarily in arguing on appeal whether the evidence at trial, viewed in the light most favorable to the person who won at trial, could cause a reasonable juror to conclude that a dating relationship was present. Those precedents would probably largely draw on cases reviewing on appeal convictions for domestic violence offenses in the states where a dating relationship is one of the elements of the crime (I strongly suspect that the language in the statute copies a definition from a state statute in some U.S. state that does just that.)
Like every definition, there would be edge cases, which is why we need judges and jurors to make these calls. But, I suspect that the early cases could involve the stereotypical heartland cases where there is little serious room to dispute the existence of a dating relationship.
Honestly, while there are lots of questions I worry about the ability of a jury of ordinary people to decide because they may not understand the legal concept that is at issue, this is not one of them.