I think it also depends on the type of child they are. I fell down the last 5 steps of the stairs ( horrible varnished slippery things) and landed at the bottom. OMG the pain that shoot up my coccyx- I screamed out and cried.
Ds8 and ds3, came running over ds8 grabbed the phone 'I'm going to call an ambulance', I was crying but reassured him it wasn't needed, so he called my dp instead (I think he needed the reassurance that it was ok not to phone 999), but it was ds3 that made me cry even more . He took hold of my hand, and he was patting my head, saying 'it's ok, little snowy is here, you will be ok mummy'.
My boys are very compassionate and empathic (ds13 came home one day from school a few months ago, and had a little cry, because his friends was being buillied- he said I know how I would feel if it was me, and that upsets me).
Some kids are a lot tougher and they don't over react, and are able to almost 'risk assess' as in, it's not serious that person will be fine. Whereas others (like mine) go into a 'it's serious' mood straight away until they know otherwise.
But neither is right or wrong. It's just how didn't kids are. Most kids will learn about empathy and compassion as they grow up, from their parents and those around them.
As it turned out, I'd actually fractured a bone in my coccyx- hence the pain! But OP I hope you are ok, it must have been a shock