I think for cuts there’s general rules for location, depth, and length to qualify as an emergency. Deep punctures/stab wounds should always be treated as emergencies, do not remove the object, but stabilize it so it doesn’t cause more damage. An amount of blood equivalent to a can of soda is life threatening.

I’ve stayed current on first aid certifications for about a decade and grew up with a parent in the medical field. I still suck at knowing when something requires an er visit, quite possibly because family, friends, and neighbors would come over and get stitches on our kitchen counter rather than go to the hospital. The best I can say is, if you are questioning whether something requires a hospital visit, then it probably does. Better to be safe than sorry.