Are there any generally accepted characteristics? For example a three finger would be a necker but a four finger wouldn't?
Nothing really generally accepted beyond "small fixed blade". (I doubt a folder in a neck sheath would count, though someone will disagree with me!)
I've seen both inverted and tip-down carry knives worn around the neck.
No reason why you couldn't wear a four-finger knife around your neck. You might not
want to for weight reasons, but it would be a necker.
I understand there are many reasons why one would own one but when you hear that someone wants one is that the first thing that generally comes to mind?
I think of woodsbumming more than self-defense. (A mercharness, that's for self-defense.)
The first thing that comes to mind for me is my Murray Carter necker. Just big enough to get a decent grip, thin and light, kydex inverted sheath. I wear it around my neck (with a break-away fastener on the paracord) or slipped into a pocket. I use it all the time in the kitchen because it's such a great slicer, and always on me. Neck carry is great when you can't necessarily get to your pockets (e.g., seated), or you're already carrying another knife that's not suitable for the current task!
I personally wouldn't carry anything bigger or heavier than my Carter, but I'm surprised at how useful it is.