The herald did stay together during the battle, but afterwards went off separately. This was not a unique circumstance of Agincourt, but typical behavior from late medieval heralds. They served an important practical functions of documenting (noble) casualties and serving as a line of communication from one camp to another. It may seem ridiculous to imagine a group of English and French officials standing together and watching their countrymen hack each other to pieces, but it wasn't as if the participation of a handful of heralds would have changed anything. By not participating in combat, they remained as an open channel between the opposing sides. Targeting heralds would be counterproductive.

The heralds did not report a victory to Henry V; Henry V summoned them personally to inform them of his victory (which of course was God's will and punishment for the transgressions of the French). It wasn't a question of determining who had won the battle. What Henry was essentially doing was making a formal statement of victory with the intent that the French heralds would go back to their own camp and communicate the English triumph to the rest of France. I suppose the closest analogy today would be a commander contacting media representatives to announce a victory. Henry wanted every Frenchman present to go back home (but especially a royal official like a herald) and tell all and sundry about the crushing defeat of a French royal army at the hands of the English. Henry is essentially saying "your king and your nobles are powerless to resist my armies, so agree to my demands or we're coming back." Of course, the French were not nearly so defeated as Henry would have hoped. While Agincourt was a major victory, the fact remained that there were plenty of French soldiers left in the country, and the English were seriously stretching their resources to embark on major campaigns like Agincourt. It wouldn't be too long before the English were practically swept off the continent in the reign of Henry VI.