From another side of it, I was sent to an incident in 2016. Job came through as "person hit by a train" with an incident location of XYZ Railway Station.
Local rag had the story online the next day, maybe the day after, as a suicide at XYZ station with a Google Streetview image of the station frontage.
Passengers seemed to be fairly satisfied with information - we weren't being bombarded with "what's going on? I need to get home!" stuff, even while we were waiting in the car park for stuff to happen before we could leave. Can't remember how many railway staff were there to assist with that. I remember it took a while for a MOM to arrive - BTP had gone ahead but we were kept back until they got there.
I would disagree with rebranding as "member of the public did something" - a person could be hit by a train by a deliberate act on their own part, an act of violence, accidental slip near the platform edge, infrastructure failure, wheelchair/pushchair caught in wind from passing train... Not really fair to paint them all with the same brush - and all of those could be "a person hit by a train" or "emergency services dealing with an incident"
Sidenote, had a colleague sent to a "person hit by train" at another station. Busier time of day. They met at an RVP with BTP, MOM, RIO, I think an ambulance team leader as well... Went to the station in convoy. Found a patient with a minor head injury - got a sliding door to the head as they were boarding/alighting. While it was technically true that the person had been hit by part of the train, the job as described did not really match the situation and the response given was inappropriate for that! Would have probably still had some delays due to "emergency services dealing with an incident", as there were emergency services, there was an incident, and I think the incident train had been instructed to wait at the station.